NBER WORKING PAPER SERIES

FIRMS AND LABOR MARKET INEQUALITY:EVIDENCE AND SOME THEORY

David CardAna Rute Cardoso

Jörg HeiningPatrick Kline

Working Paper 22850http://www.nber.org/papers/w22850

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138November 2016

We are extremely grateful to Raffaele Saggio for assistance in preparing this paper, and to Katherine Shaw and David Green for helpful suggestions on an earlier draft. Cardoso acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (Severo Ochoa Programme for Centres of Excellence in R&D grant SEV-2015-0563) and the Research Council of Norway (Europe in Transition funding scheme project 227072/F10 at ESOP). The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

© 2016 by David Card, Ana Rute Cardoso, Jörg Heining, and Patrick Kline. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.

Firms and Labor Market Inequality: Evidence and Some TheoryDavid Card, Ana Rute Cardoso, Jörg Heining, and Patrick KlineNBER Working Paper No. 22850November 2016JEL No. D24,J31,J42

ABSTRACT

We survey two growing bodies of research on firm-level drivers of labor market inequality. The first examines how wages are affected by differences in employer productivity. Studies that focus on firm-specific productivity shocks and control for the non-random sorting of workers to firms typically find that a 10% increase in value-added per worker leads to somewhere between a 0.5% and 1.5% increase in wages. Given the wide variation in firm-specific productivity, elasticities of this size suggest that a significant fraction of wage inequality is tied to firm performance. A second literature estimates two-way fixed effects models that rely on the wage changes of people who move between firms to identify firm-specific wage premiums. This literature also concludes that firm pay setting is important for wage inequality, with many studies finding that firm wage effects contribute approximately 20% of the overall variance of wages. To interpret these findings, we develop a model of firm wage setting in which workers have idiosyncratic tastes for different workplaces. We show that simple versions of this model can rationalize the standard two-way fixed effects specification proposed by Abowd, Kramarz and Margolis (1999), and can also match the typical “rent-sharing” elasticities estimated in the literature. Extended versions of the model can potentially explain differences in the wage premiums paid by a given employer to different subgroups of workers.

David CardDepartment of Economics549 Evans Hall, #3880University of California, BerkeleyBerkeley, CA 94720-3880and NBERcard@econ.berkeley.edu

Ana Rute CardosoIAE (CSIC) and Barcelona GSE08193 BellaterraBarcelonaSpainanarute.cardoso@iae.csic.es

Jörg HeiningInstitute for Employment Research (IAB)Joerg.Heining@iab.de

Patrick KlineDepartment of EconomicsUniversity of California, Berkeley530 Evans Hall #3880Berkeley, CA 94720and NBERpkline@econ.berkeley.edu

How much does where you work determine what you earn? In the standard competitivelabor market model firms take market wages as given and firm-specific heterogeneity influ-ences who is hired, but not the level of pay of any particular worker. The pervasive influenceof this perspective is evident in major reviews of the wage inequality literature (Katz andAutor, 1999; Goldin and Katz, 2009; Acemoglu and Autor, 2011), which focus almost exclu-sively on the role of market-level skill prices in driving inequality trends.1 This view standsin stark contrast to the Industrial Organization literature, which typically models marketsas imperfectly competitive (Tirole, 1988; Pakes, 2016). Though economists seem to agreethat part of the variation in the prices of cars and breakfast cereal is due to factors otherthan marginal cost, the notion that wages differ substantially among equally skilled workersremains highly controversial.

A long tradition in labor economics posits that employers have significant latitude toset wages (e.g., Robinson, 1933; Reynolds, 1946; Lester, 1946; Slichter, 1950), a view whichfound early support in empirical studies of industry wage differentials (Katz, 1986; Kruegerand Summers, 1988; Katz and Summers, 1989). Yet it has proven difficult to convincinglydistinguish industry wage premia from subtle forms of dynamic sorting (Murphy and Topel,1990; Gibbons and Katz, 1992; Gibbons et al., 2005). The growing availability of matchedemployer-employee datasets offers opportunities to study these questions at the more gran-ular level of the firm. Nevertheless, many of the fundamental identification problems thatplagued earlier studies carry over to these new datasets. This review summarizes what hasbeen learned so far from these new datasets about the importance of firms in wage setting,and what challenges remain.

Our starting point is the widely accepted finding that observably similar firms exhibitmassive heterogeneity in measured productivity (e.g., Syverson, 2011). A natural question iswhether some of these productivity differences spill over to wages. The prima facie case forsuch a link seems quite strong: a number of recent studies find that trends in aggregate wagedispersion closely track trends in the dispersion of productivity across workplaces (Dunneet al., 2004; Faggio, Salvanes, and Van Reenen, 2010; Barth et al. 2016). However, theseaggregate relationships are potentially driven in part by changes in the degree to whichdifferent groups of workers are assigned to different firms.

Two distinct literatures attempt to circumvent the sorting issue using linked employer-employee data. The first literature studies the impact of shocks to firm productivity onthe wages of workers. The resulting estimates are typically expressed as “rent-sharing” elas-

1This market-wide perspective is also common in economic models of discrimination, which typicallyhave no role for firm-specific factors to affect the wages of female or minority workers (see e.g., Charles andGuryan, 2008, 2011).

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ticities. A review of this literature suggests that estimated rent sharing elasticities aresurprisingly robust to the choice of productivity measure and labor market environment:most studies that control for worker heterogeneity find wage-productivity elasticities in therange 0.05-0.15, though a few older studies find larger elasticities. We also provide some newevidence on the relationship between wages and firm-specific productivity using matchedworker-firm data from Portugal. We investigate a number of specification issues that fre-quently arise in this literature, including the impact of filtering out industry-wide shocks,different approaches to measuring rents, and econometric techniques for dealing with unob-served worker heterogeneity.

A second literature which builds on the additive worker and firm effects wage modelproposed by Abowd, Kramarz, and Margolis (1999) (hereafter AKM), uses data on wageoutcomes as workers move between firms to estimate firm-specific pay premiums. Thisliterature also finds that firms play an important role in wage determination, with a typicalfinding that about 20% of the variance of wages is attributable to stable firm wage effects. Wediscuss some of the issues that arise in implementing AKM’s two-way fixed effects estimator,which is the main tool used in this literature, and evidence on the validity of the assumptionsunderlying the AKM specification.

We then attempt to forge a more direct link between the rent sharing literature andstudies based on the AKM framework. Using data from Portugal we show that more pro-ductive firms pay higher average wage premiums but also tend to hire more productiveworkers. Indeed, we estimate that about 40% of the observed difference in average hourlywages between more and less productive firms is attributable to the differential sorting ofhigher-ability workers to more productive firms, underscoring the importance of controllingfor worker heterogeneity. Consistent with the additive specification underlying the AKMmodel, we find that the wage premiums offered by more productive firms to more- and less-educated workers are very similar, and that the relative wage of highly educated workers isremarkably stable across firms.

In the final section of the paper we develop a stylized model of imperfect competitionin the labor market that provides a tractable framework for studying the implications ofworker and firm heterogeneity for wage inequality. Our analysis builds on the static partialequilibrium monopsony framework introduced by Joan Robinson (1933) which, as noted byManning (2011), captures many of the same economic forces as search models, albeit withoutproviding a theory of worker flows between jobs. We provide a microeconomic foundation forimperfect labor market competition by allowing workers to have heterogeneous preferencesover the work environments of different potential employers.2 This “workplace differentiation”

2In their review of monopsony models, Boal and Ransom (1997) refer to this as the case of “classic

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can reflect heterogeneity in firm location, job characteristics (e.g., corporate culture, startingtimes for work), or other factors that are valued differently by different workers. Regardlessof its source, such heterogeneity makes employers imperfect substitutes in the eyes of workers,which in turn gives firms some wage-setting power.

To capture this heterogeneity, we adopt a random utility model of worker preferences thatcharacterizes firm-specific labor supply functions. We presume, as in Robinson’s analysis andmuch of the Industrial Organization literature, that the firm cannot price discriminate basedupon a worker’s idiosyncratic preference for the firm’s work environment. Hence, rather thanoffer each worker her reservation wage (e.g., as in Postel-Vinay and Robin, 2002), firms posta common wage for each skill group that is marked down from marginal product in inverseproportion to their elasticity of labor supply to the firm. This condition provides a naturalanalogue to the equilibrium markups of price over marginal cost found in workhorse modelsof differentiated product demand and pricing (Berry, 1994; Berry, Levinsohn, and Pakes,1995; Pakes, 2016).

We show that many well-documented empirical regularities can be rationalized in thisframework. Firm heterogeneity in productivity affects not only the firm size distribution,but also the distribution of firm-specific wage premiums and the degree of sorting of differentskill groups across firms. Calibrating a simple version of the model to observed rent sharingestimates yields predicted wage dispersion in excess of what has been found in a wide class ofsearch models (Hornstein, Krussell, Violante, 2011). We also present conditions under whichlog wages are additively separable into worker and firm components, as in the pioneeringeconometric model of AKM. Specifically, we show that the firm-specific wage premium willbe constant across skill groups if they are perfect substitutes in production and if differentskill groups have a similar relative valuation for wages versus non-wage components in theirassessment of alternative jobs. Even under these conditions, however, the market-level wagegap between skill groups will reflect differences in their employment distributions across moreand less productive firms. More generally, groups that put a higher relative value on wagesin comparing different jobs will receive a larger wage premium at more profitable firms.

We conclude with some thoughts on unresolved empirical and theoretical issues in theliterature. An important challenge for empirical work on rent sharing is the identification offirm-specific shocks to productivity, as correlated shocks can yield market level wage adjust-ments. In the firm switching literature a key question is whether conventionally estimatedfirm-specific pay premiums predict the wage changes associated with exogenously inducedjob accessions and separations. On the theoretical side, important questions include howto model strategic labor market interactions between firms and to what extent the insights

differentiation”.

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from simple static wage setting models of workplace differentiation carry over to dynamiclabor market settings with search or mobility frictions.

1 Productivity, wages, and rent sharing

A large empirical literature reviewed by Syverson (2011) documents that firms, like workers,exhibit vast heterogeneity in productivity. For example, Syverson (2004) finds that the 90thand 10th percentiles of total factor productivity (TFP ) among US manufacturing firmsdiffer by an average factor of approximately two within 4-digit industries. Hsieh and Klenow(2009) find even larger productivity gaps in India and China, with 90-10 TFP ratios onthe order of five. While the variation in measured productivity probably overstates the trueheterogeneity in plant-level efficiency, there is also strong evidence in the literature thatmeasured productivity conveys real information. For example, measured TFP is a strongpredictor of firm survival (Foster, Haltiwanger, and Syverson, 2008).

It is natural to wonder if these large productivity differences lead to differences in workerpay. In fact, an extensive literature has documented the existence of substantial wage differ-ences across plants and establishments (Lester, 1946; Slichter, 1950; Davis and Haltiwanger,1991; Groshen, 1991; Bernard and Jensen, 1995; Cardoso, 1997; Cardoso, 1999; Skans, Edin,and Holmlund, 2009; Song et al., 2015) that are strongly correlated with basic measures ofproductivity. Nevertheless, economists have been reluctant to interpret these differences aswage premiums or rents, since it has been difficult to know how unobserved worker qualitydiffers across plants.

Recent studies, however, have documented some striking links between establishmentlevel productivity and wage dispersion (Dunne et al., 2004; Faggio, Salvanes, and VanReenen, 2010; Barth et al. 2016). Figure 1 plots results from Barth et al. (2016), showingremarkably similar trends in the dispersion of wages and productivity across business estab-lishments in the United States. Taken at face value, these parallel trends are consistent witha roughly unit elasticity of establishment wages with respect to productivity (see Barth et al.,2016, p. S71). Of course, Figure 1 does not tell us whether the composition of the workforceemployed at these establishments is changing over time. What appear to be more productiveestablishments may simply be establishments that hire more skilled workers, which is fullyconsistent with the competitive labor market model in which all firms pay the same wagesfor any given worker.

A more direct attack on the question of whether firm-specific productivity differentialsfeed into differences in wages comes from the empirical literature on “rent-sharing.” AppendixTable 1 describes 22 recent studies in this literature. The basic idea in these papers is to

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relate wages to some measure of employer profitability or rents. This is in many respects,the analogue of a large literature in international economics and industrial organization onthe pass-through of cost shocks to prices. A typical finding in that literature is that shocksto marginal cost (typically measured via exchange rate fluctuations) yield muted fluctua-tions in product prices, signaling the presence of market power (Goldberg and Hellerstein,2013; Gorodnichenko and Talavera, forthcoming). Most rent sharing studies find similarlyimperfect propagation of productivity shocks to wages. However, the elasticities reported inthis literature are often sensitive to measurement issues, which we now review.

Measuring rents

The empirical rent sharing literature is motivated by an assumed structural relationshipbetween wages and either profit per worker or a measure of “quasi-rent” per worker. Tofacilitate discussion, suppose that there is a single type of labor at a firm j, and that thewage (wj) is determined by a structural relationship of the form:

wj = b+ γQj/N j (1)

where b represents an alternative wage, Nj is employment at the firm, Qj represents quasi-rents, and γ is a rent-sharing parameter.3 The firm combines labor inputs and capital (Kj)

and faces an exogenous rental rate r on capital, yielding the quasi-rent:

Qj = V Aj − bNj − rKj

where V Aj is value added (revenue net of materials costs). Value added is related to laborand capital inputs by:

V Aj = PjTjf(N j, Kj)

where Pj is a potentially firm-specific selling price index, Tj is an index of technical efficiency,and f is a standard production function. Here PjTj represents total factor productivity(TFPj) which, in the terminology of Foster, Haltiwanger and Syverson (2008), is also referredto as “revenue productivity” because it is the product of “physical productivity” Tj andproduct price Pj.

3 This equation can be derived by assuming that workers as a group bargain with the firm to maximizea weighted product of profits and the excess wage bill Nj(wj − b), where the weight on the wage bill isγ. See e.g., de Menil (1971). Strictly speaking this derivation assumes that Q/N is exogenous to the levelof wages, as is the case when employment and capital are efficiently determined by joint bargaining. SeeSvejnar (1986) and Card, Devicienti, and Maida (2014) for a related discussion of potential holdup problemsin the determination of capital.

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We assume that TFPj is the driving source of variation that researchers are implicitlytrying to model in the rent sharing literature. Under this interpretation, firm-specific TFPshocks lead to changes in quasi-rent per worker that cause wages to fall or rise relative to thealternative wage. The elasticity of wages with respect to an exogenous change in quasi-rentper worker is:

ξQj =γQjNj

bj + γQjNj

(2)

which corresponds to the share of rents in wages. The elasticity of wages with respect toprofit per worker (πj/Nj) should be of comparable magnitude. Indeed, under the usualbargaining interpretation of equation (1) profits per worker are a constant share of quasi-rents per worker:

πjNj

= (1− γ)Qj

Nj

.

Rather than measure quasi-rents, a majority of studies relate wages to value-added perworker. The elasticity of wages with respect to value added per worker is

ξj = ξQj ×V AjQj

which will be bigger than ξQj, since Qj < V Aj.4 For example, data reported in Card,Devicienti, and Maida (2014) suggests that the ratio of value added to quasi-rent for firms inNortheastern Italy is typically around 2. Finally, some studies in the literature relate wagesto revenue per worker, rather than to value added per worker. Under the assumption thatintermediate input costs represent a constant share of revenues the elasticity of wages withrespect to revenue per worker will be equal to the elasticity with respect to value added perworker. More generally if intermediate input costs vary – for example because of varyingenergy costs – and these costs are passed through to the firm’s consumers, one would expecta smaller elasticity of wages with respect to revenues per worker.

An important confounding factor in the rent sharing literature is variation in workerquality. Firms that employ more highly skilled workers would be expected to have higherrevenue per worker and value added per worker, and also pay higher wages, leading to apotential upward bias in the measured rent sharing elasticity in cross-sectional studies thatcompare different firms at a point in time. A similar bias can also arise in longitudinal studiesthat compare changes in firm-specific wages and profitability over time if there are unobserved

4Again, this is only true under the assumption that the levels of value-added and quasi-rents are exoge-nously determined, as in an efficient bargaining model. More generally, the ratio of value-added per workerto quasi-rent per worker can depend on the wage.

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changes in the skill characteristics of workers. For this reason, when we summarize the studiesin the literature we classify studies based upon whether the research design includes controlsfor unobserved worker skills.

A summary of the rent sharing literature

Table 1 synthesizes the estimated rent sharing elasticities from the 22 studies listed in Ap-pendix Table 1, extracting one or two preferred specifications from each study and adjustingall elasticities to an approximate value-added-per-worker basis.5 We divide the studies intothree broad generations based on the level of aggregation in the measures of rents and wages.

The first group of studies, which includes two influential papers from the early 1990s,uses industry-wide measures of productivity and either individual-level or firm-wide averagewages. The average rent sharing elasticity in this group is 0.16. A second generation ofstudies includes five papers, mostly from the mid-1990s, that use firm- or establishment-specific measures of rents but measure average wages of employees at the workplace level.The average rent sharing elasticity in this group is 0.15, though there is a relatively widerange of variation across the studies. Given the likely problems caused by variation in workerquality, we suspect that most first generation and second generation studies yield upward-biased estimates of the rent sharing elasticity.

A third generation of studies consists of 18 relatively recent papers that study the link be-tween firm- or establishment-specific measures of rents and individual-specific wages. Manyof these studies attempt to control for variation in worker quality, in some cases by studyingthe effect of changes in measured rents on changes in wages. In this group the mean rentsharing elasticity is 0.08, though a few studies report rent sharing elasticities that are 0.05 orsmaller. To quantify the potential role of rent-sharing in the overall dispersion of wages, notethat the standard deviation of average value added per worker across firms in the Portuguesedata we analyze in the next section is about 0.70, with a spead between the 90th and 10thpercentiles of around 1.6. An elasticity of 0.08 implies that the variation in productivityacross firms creates a “Lester range” of wage variability (Lester, 1952) of about 13 log pointsbetween firms at the 90th and 10th percentiles.

Although significant progress has been made in this literature, none of these studies isentirely satisfactory. Very few studies have clear exogenous sources of variation in produc-tivity. Most papers (e.g., Card, Cardoso, and Kline, 2016; Carlsson, Messina, and Skans,2014; Guiso, Pistaferri, and Schivardi, 2005) rely on timing assumptions about the stochasticprocess driving productivity to justify using lags as instruments. A notable exception is Van

5We extract an IV estimate when one is available, and convert elasticities with respect to profit per workeror quasi-rent per worker to a value added per worker basis by multiplying by 2.

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Reenen (1996), who studies the effects of major firm innovations on employee wages. Hefinds a very large rent sharing elasticity of 0.29 but this figure may be upward biased byskill upgrading on the part of innovative firms – a concern he could not address with ag-gregate data. Other studies (e.g., Abowd and Lemieux, 1993; Card, Devicienti, and Maida,2014) use industry level shocks as instruments for productivity. However, these instrumentsmay violate the exclusion restriction if labor supply to the sector is inelastic since evenfully competitive models predict that industry level shocks can yield equilibrium wage re-sponses. Moreover, industry level shocks might yield general equilibrium responses thatchange worker’s outside options (Beaudry, Green, and Sand, 2012). Finally, with the moveto matched employer-employee microdata, economists have had to contend with serious mea-surement error problems that emerge when constructing plant level productivity measures.It remains to be seen whether instrumenting using lags fully resolves these issues.

Specification issues: a replication in Portuguese data

To supplement the estimates in the literature and probe the impact of different design choiceson the magnitude of the resulting elasticities we conducted our own analysis of rent sharingeffects using matched employer-employee data from Portugal. The wage data for this exercisecome from Quadros de Pessoal (QP), a census of private sector employees conducted eachOctober by the Portuguese Ministry of Employment. We merge these data to firm-specificfinancial information from SABI (Sistema de Analisis de Balances Ibericos) database, dis-tributed by Bureau van Dijk.6 We select all male employees observed between 2005 and 2009who work in a given year at a firm in the SABI database with valid information on sales perworker for each year from 2004 to 2010, and on value added per worker for each year from2005 to 2009.

Panel A of Table 2 presents a series of specifications in which we relate the log hourlywage observed for a worker in a given year (between 2005 and 2009) to mean log value addedper worker or mean log sales per worker at his employer, averaged over the sample period.These are simple cross-sectional rent sharing models in which we use an averaged measureof rents at the employer to smooth out the transitory fluctuations and measurement errorsin the financial data. In row 1 we present models using mean log value added per worker asthe measure of rents; in row 2 we use mean log sales per worker; and in row 3 we use mean

6Businesses in Portugal are required to file income statements and balance sheet information annually attheir local Commercial Registry (Conservatoria do Registro Comercial). These reports are publicly accessibleand are collected by financial service firms and assembled into the SABI database. We merge SABI and QPusing information on detailed location, industry, firm creation date, shareholder equity, and annual sales thatare available in both data sets. See Card, Cardoso and Kline (2016) for more information on the matchingprocess.

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log value added per worker over the 2005-2009 period but instrument this with mean logsales per worker over a slightly wider window (2004-2010). For each choice we show a basicspecification (with only basic human capital controls) in column 1, a richer specification withcontrols for major industry and city in column 2, and a full specification with dummies for202 detailed industries and 29 regions in column 3.

Two main conclusions emerge from these simple models. First, the rent sharing elasticityis systematically larger when rents are measured by value added per worker than by salesper worker.7 Second, the rent sharing elasticities from this approach are relatively large.Interestingly, the 0.20 to 0.30 range of estimates is comparable to the range of the studies inthe first two panels of Table 1.

An obvious concern with the specifications used in Panel A is that they fail to fully controlfor variation in worker quality. As discussed above, this is likely to lead to an upward biasin the relationship between wages and value added per worker. The specifications in PanelB of Table 2 partially address this by examining the effect of changes in firm specific rentson changes in wages for workers who remain at the firm over the period from 2005 to 2009– a within-job or “stayers” design. We present three sets of specifications of this design.The models in row 4 measure the change in rents by the change in log value added perworker. The models in row 5 use the change in log sales per worker. The models in row 6use the change in value added per worker as the measure of rents, but instrument the changeusing the change in sales per worker over a slightly wider interval to reduce the impact ofmeasurement errors in value added.8

Relative to the cross-sectional models, the within-job models yield substantially smallerrent sharing elasticities. This difference is likely due to some combination of unobservedworker quality in the cross-sectional designs (which leads to an upward bias in these speci-fications), measurement error (which causes a larger downward bias in the stayer designs),and the fact that value added fluctuations may include a transitory component that firmsinsure workers against (Guiso, Pistaferri, and Schivardi, 2005).9 The discrepancy is particu-larly large for OLS models using sales per worker (compare row 2 and row 5 of Table 2): theelasticity for stayers is only about one-tenth as large as the cross-sectional elasticity. We sus-pect that measurement errors and transitory fluctuations in annual sales are relatively large,

7A similar finding is reported by Card, Devicienti, and Maida (2014) using Italian data.8If measurement errors in value added per worker in year t are uncorrelated with errors or fluctuations in

sales per worker in years t+1 and t− 1, then the use of a “bracketing” instrument will eliminate the effect ofmeasurement error in value added. We suspect that this is only partially true, so the IV approach reducesbut does not fully eliminate the effect of errors in value added.

9A third potential explanation is selection bias in the stayer models, induced by selecting a sample of jobstayers. Results presented in Card, Cardoso and Kline (2016, Appendix Table B10) suggest this factor isrelatively small.

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and the impact of these factors is substantially magnified in the within-job specificationsestimated by OLS. Given the presence of errors and idiosyncratic fluctuations, we prefer theIV estimates in row 6, which point toward a rent sharing elasticity of approximately 0.06.

An interesting feature of both the OLS and IV within-job estimates is that the additionof detailed industry controls reduces the rent sharing elasticity by 10-20 percent. Since theseindustry dummies absorb industry-wide productivity shocks that are shared by the firmsin the same sector, we conclude that the rent sharing elasticity with respect to firm-specificproductivity shocks (which is estimated by the models in column 3) is somewhat smaller thanthe elasticity with respect to sector-wide shocks (which are incorporated in the elasticities inthe models in column 1). If true more generally, this suggests that the use of industry-widerent measures will lead to a somewhat larger rent sharing elasticities than would be obtainedusing firm-specific productivity measures and controlling for industry-wide trends. A similarconclusion is reported by Carlsson, Messina, and Skans (2014).

Overall, we conclude that the estimated elasticities of wages with respect to value addedper worker in Portugal are comparable in magnitude to the estimates in the existing rent-sharing literature summarized in Table 1. A typical estimate in our data and the literaturefrom specifications that control for worker quality is between 0.05 and 0.10, though thereare a few estimates on each side of this range.

2 Firm switching

While the rent-sharing literature documents a strong correlation between firm profitabilityand pay, a parallel literature finds that workers who move between firms (or establishments)experience wage gains or losses that are highly predictable. In this section we provide anoverview of recent findings from this approach and discuss some of the major issues inthis literature. In the following section we discuss how the firm-specific wage premiumsestimated by studies of firm switching are related to measures of firm profitability, providinga link between the rent sharing and firm switching literatures.

AKM models

In their seminal study of the French labor market, AKM specified a model for log wages thatincludes additive effects for workers and firms. Specifically, their model for the log wage ofperson i in year t takes the form:

lnwit = αi + ψJ(i,t) +X ′itβ + εit

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where Xit is a vector of time varying controls (e.g., year effects and controls for experience),αi is a “person effect” capturing the (time-invariant) portable component of earnings abil-ity, the {ψj}Jj=1 are firm-specific relative pay premiums, J (i, t) is a function indicating theemployer of worker i in year t, and εit is an unobserved time-varying error capturing shocksto human capital, person-specific job match effects, and other factors. The innovation inAKM’s framework is the presence of the firm effects, which allow for the possibility thatsome firms pay systematically higher or lower wages than other firms. Specifically, the AKMmodel predicts that workers who move from firm k to firm j will experience an average wagechange of ψj−ψk, while those who move in the opposite direction will experience an averagechange of ψk − ψj – a stark “symmetry” prediction that we discuss in more detail below.

Estimates of AKM style models on population level administrative datasets from a varietyof different countries have found that the firm effects in these models typically explain 15-25 percent of the variance of wages – less than the person effects, but enough to indicatethat firm-specific wage setting is important for wage inequality.10 One problem with thisassessment is that the person and firm effects are estimated with considerable imprecision,which means the explanatory power of firms will typically be somewhat overstated – aproblem that was also recognized in the earlier literature on industry wage differentials(Krueger and Summers, 1988). Andrews et al. (2008) provide an approach to dealing withthis problem that we discuss in more detail below.

If different firms pay different wage premiums, the pattern of sorting of workers to firmswill also matter for overall wage inequality. In particular, the variance of log wages is:

V ar (lnwit) = V ar (αi) + V ar(ψJ(i,t)

)+ V ar (X ′itβ) + V ar (εit) (3)

+2Cov(αi, ψJ(i,t)

)+ 2Cov (αi, X

′itβ) + 2Cov

(ψJ(i,t), X

′itβ)

which includes both the variance of the firm-specific wage premiums and a term reflectingthe covariance of the worker and firm effects. If workers with higher earning capacity aremore likely to work at higher-premium firms, then this covariance term will be positive, andany inequality effects from the presence of the firm premiums will be amplified.

10For example, Abowd, Lengermann, and McKinney (2003) find that firm effects comprise 17% of thevariance of US wages. Card, Heining, and Kline (2013) find that establishment effects explain between 18%and 21% of the variance of the wages of German men depending on the time period studied. Card, Cardoso,and Kline (2016) find that firm effects explain 20% of the variance of hourly wages for Portuguese men and17% of the variance for women. Macis and Schivardi (2016) find that firm effects explain 15% of the wagevariance of Italian manufacturing workers. Finally, Lavetti and Schmutte (2016) find that establishmenteffects explain 21% of the variance of wages of workers in the formal sector in Brazil.

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An alternative decomposition uses the fact that:

V ar (lnwit) = Cov (lnwit, αi)+Cov(lnwit, ψJ(i,t)

)+Cov (lnwit, X

′itβ)+Cov (lnwit, εit) . (4)

This yields an “ensemble” assessment of the importance of each variance component to wagedispersion that includes the contribution of the covariance between wage components. Forexample, under this decomposition, the contribution of the firm component to total wagevariation would be Cov

(lnwit, ψJ(i,t)

)= V ar

(ψJ(i,t)

)+Cov

(αi, ψJ(i,t)

)+Cov

(X ′itβ, ψJ(i,t)

).

One way to think about this decomposition is that one half of the firm covariance terms in(3) are attributed to the firm-specific wage premiums.

Identifying age and time effects

A technical issue that arises with the AKM model is appropriate specification of the effectsof age. Following Mincer (1974), it is conventional to include a polynomial in age or potentialexperience (age minus education minus 6) in Xit. However, it is also standard to include aset of year indicators in Xit to adjust for changing macroeconomic conditions. This raises anidentification problem because age (ait) can be computed as calendar year (t) minus birthyear (bi). Hence, we face the classic problem of distinguishing additive age, year, and cohorteffects, where cohort effects are understood to load into the person effects.

In their original paper, AKM solved this problem by using “actual” labor market expe-rience (i.e. the number of years the worker had positive earnings since entering the labormarket) which, if some employment histories have gaps, will not be perfectly collinear withyear and person dummies. While in some respects this provides a simple fix to the problem,there are two important drawbacks. First, it is not always possible to reconstruct a worker’semployment history, both because some datasets do not always go far enough back to coverthe cohorts of interest and because some datasets only report point in time measures ofemployment (e.g. who was on the payroll in October) rather than a complete history of allemployment spells in all years. Second, it is not clear that employment gaps are exogenous,even conditional on a person effect. For example, leaving employment for an entire year couldreflect severe health shocks that directly influence earnings ability and confound estimationof relative firm pay.

An alternative approach to dealing with this problem is to impose a linear restrictionon the effects of age or time. While the firm effects are invariant to how age and timeeffects are normalized, different normalizations will yield different values of the person effectsand the covariate index X ′itβ. Card, Heining, and Kline (2013) allow for separate thirdorder polynomials in age by education group along with unrestricted year effects. To obtain

13

identification, they restrict the age profile to be flat at age 40. This is accomplished byomitting the linear age term for each education group and using a cubic polynomial in (ait−40). The same restriction is used in Card, Cardoso, and Kline (2016). While this restrictionis unlikely to hold exactly, there is reason to believe it provides a good approximation to theshape of the age-earnings profile.11

Table 3 examines the sensitivity of the results in Card, Cardoso, and Kline (2016) tofour alternate normalizations of the age effects. The first column shows the baseline normal-ization, which attributes a relatively small fraction of the overall variance of wages to thetime-varying individual component of wages. Renormalizing the age profile to be flat at age50 (column 2) has little effect on this conclusion, whereas re-normalizing the profile to beflat at age 30 leads to a slightly larger variance share for the time-varying component, andalso implies a relatively strong negative correlation between the person effects and the indexX ′itβ. Normalizing the age profile to be flat at age 0 – which is what is being done by simplyomitting the linear term from an uncentered age polynomial – exacerbates this pattern andleads to a decomposition that suggests that the variances of αi and X ′itβ are both very largeand that the two components are strongly negatively correlated.12 Figure 2 contrasts theimplied age profiles for four single year-of-birth cohorts of low-education men from this naivespecification with the implied profiles for the same groups under the baseline normalization.Evidently, the strong negative correlation between the person effects and the covariate indexreported in column 4 of Table 3 is driven by implausibly large cohort effects, which trend ina way to offset the imposed assumption that the cubic age profile is flat at age 0.

Rather than restricting the age profile to be flat at a point, we can also achieve identifi-cation by assuming the true profile is everywhere nonlinear. Column 5 shows the results ofusing a linear combination of normal density functions in age (with five year bandwidths)to approximate the age profile.13 Because each Gaussian component is nonlinear, we do notneed restrictions on the parameters to avoid collinearity with cohort and time effects. Nev-ertheless, using Gaussian basis functions will only “solve” the identification problem if thetrue age profile has no linear segments. As shown in column 5, the Gaussian approximationyields results somewhere between our baseline normalization and the specification in column3: although the estimated variability of the worker, firm, and time varying components isvery close to baseline, the correlation of the person effects and X ′itβ becomes slightly nega-

11For example, as shown in Figures 3a-3c of Card and Cardoso (2012) the age profile of wages for Portuguesemen tends to be relatively flat after age 40.

12Abowd, Lengermann and McKinney (2003) impose a normalization on the experience profiles in theirestimation of an AKM model for the LEHD data that leads to large variances of the αi and X ′itβ components,and a large negative covariance (ρ = −0.55), similar to the pattern in column 4.

13Letting n (.) denote the standard normal density, we use basis functions of the form n(ait−x

5

)where

x ∈ {20, 25, ..., 65}.

14

tive. Fortunately, the covariance of the person and firm effects is essentially the same underour baseline normalization and the Gaussian specification, leading us to conclude that mostof the statistics of interest in this literature found under an age 40 normalization are robustto alternate identifying assumptions.

To summarize: in comparing results from different applications of the AKM frameworkresearchers should pay close attention to the choice of normalization. The values of the personeffects (i.e., the αi’s) and the time varying controls (i.e., X ′itβ) are not separately identifiedwhen Xit includes both year effects and a linear age term. The choice of normalization hasno effect on estimates of V ar(ψJ(i,t)), V ar(αi+Xitβ), or the covariance term Cov(ψJ(i,t), αi+

Xitβ) but, as shown in Table 3, it will affect the estimated covariance of the person and firmeffects and the relative magnitudes of V ar(αi) and V ar(ψJ(i,t)).

Worker-firm sorting and limited mobility bias

In their original study, AKM reported a negative correlation between the estimated workerand firm effects, suggesting that sorting of workers to different firms tended to reduce ratherthan increase overall wage inequality. Subsequent research, however, has typically foundpositive correlations. For example, Abowd, Lengermann, and McKinney (2003) report acorrelation of 0.08 for U.S. workers, while Card, Heining and Kline (2013) report a correlationof 0.23 for male German workers in the 2000s. As discussed by Abowd et al. (2004) andAndrews et al. (2008) these correlations are biased down in finite samples with the sizeof the bias depending inversely on the degree of worker mobility among firms. Maré andHyslop (2006) and Andrews et al. (2012) show convincingly that this “limited-mobility” biascan be substantial. In sampling experiments they find that the correlation of the estimatedeffects becomes more negative when the AKM model is estimated on smaller subsets ofthe available data. While Andrews et al. (2008) and Gaure (2014) provide approaches tocorrecting for this downward bias in the correlation (and the upward biases in the estimatedvariances of person and firm effects), their procedures require a complete specification of thecovariance structure of the time-varying errors, which makes such corrections highly modeldependent.14 The development of corrections that are robust to unmodeled dependence andheteroscedasticity is an important priority for future research.

14For example Andrews et al. (2008) compute bias corrections in a linked sample of German workersand establishments under the assumption that the transitory errors in wages are homoscedastic and seriallyuncorrelated. They find that the corrections have little effect on the estimated correlation between workerand firm effects. However, subsequent results by Andrews et al. (2012) show large biases in the estimatedcorrelation when the AKM model is estimated on subsamples as large as 30% of the data.

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Exogenous mobility

AKM’s additive worker and firm effect specification is simple and tractable. Nevertheless, ithas been widely criticized because OLS estimates of worker and firm effects will be biasedunless worker mobility is uncorrelated with the time-varying residual components of wages.In an attempt to provide some transparent evidence on this issue, Card, Heining, and Kline(2013) (hereafter, CHK) develop a simple event-study analysis of the wage changes experi-enced by workers moving between different groups of firms. Rather than rely on a model-based grouping, CHK define firm groups based on the average pay of coworkers. If the AKMmodel is correct and firms offer proportional wage premiums for all their employees, thenworkers who move to firms with more highly-paid coworkers will on average experience payraises, while those who move in the opposite direction will experience pay cuts. Moreover,the gains and losses for movers in opposite directions between any two groups of firms willbe symmetric. In contrast, models of mobility linked to the worker-and-firm-specific matchcomponent of wages (e.g., Eeckhout, and Kircher, 2011) imply that movers will tend to expe-rience positive wage gains regardless of the direction of their move, violating the symmetryprediction.

Figures 3 and 4 present the results of this analysis using data for male and female workersin Portugal, taken from Card, Cardoso, and Kline (2016). The samples are restricted toworkers who switch establishments and have at least two years of tenure at both the originand destination firm. Firms are grouped into coworker pay quartiles (using data on maleand female coworkers). For clarity, only the wage profiles of workers who move from jobs inquartile 1 and quartile 4 are shown in the figures. The wage profiles exhibit clear step-likepatterns: when workers move to higher paying establishments their wages rise; when theymove to lower paying establishments their wages fall. For example, males who start at a firmin the lowest quartile group and move to a firm in the top quartile have average wage gainsof 39 log points, while those who move in the opposite direction have average wage lossesof 43 log points. The gains and losses for other matched pairs of moves are also roughlysymmetric, while the wage changes for people who stay in the same coworker pay group areclose to zero.

Another important feature of the wage profiles in Figures 3 and 4 is that wages of thevarious groups are all relatively stable in the years before and after a job move. Workerswho are about to experience a major wage loss by moving to a firm in a lower coworkerpay group show no obvious trend in wages beforehand. Similarly, workers who are about toexperience a major wage gain by moving to a firm in a higher pay group show no evidenceof a pre-trend. By contrast, if worker mobility were driven by gradual employer learning,we would expect wage changes to precede moves between firm quality groups over the time

16

horizons examined (Lange, 2007).Card, Cardoso, and Kline (2016) also present simple tests of the symmetry restrictions

imposed by the AKM specification, using regression-adjusted wage changes of males andfemales moving between firms in the 4 coworker pay groups. Comparisons of upward anddownward movers are displayed visually in Figures 5a and 5b, and show that the matchedpairs of adjusted wage changes are roughly scattered along a line with slope -1, consistentwith the symmetry restriction.

Similar figures can be constructed using firm groupings based on the estimated pay effectsobtained from an AKM model. As shown in CHK (Figure VII), applying this approach todata for German males yields the same conclusions as an analysis based on coworker paygroups. Macis and Schivardi (2016) report such diagnostics using Social Security earningsdata for Italian workers, and confirm that wage profiles of movers exhibit the same step-likepatterns found in Germany and Portugal.

Additive separability

Another concern with the AKM model is that it presumes common proportional firm wageeffects for all workers. One way to evaluate the empirical plausibility of the additive AKMspecification is to examine the pattern of mean residuals for different groups of workersand firms. Figure 6 and 7, taken from Card, Cardoso and Kline (2016) shows the meanresiduals for 100 cells based on deciles of the estimated worker effects and deciles of theestimated firm effects. If the additive model is correct, the residuals should have mean zerofor matches composed of any grouping of worker and firm effects, while if the firm effectsvary systematically with worker skill we expect departures from zero. Reassuringly, the meanresiduals are all relatively close to zero. In particular, there is no evidence that the mostable workers (in the 10th decile of the distribution of estimated person effects) earn higherpremiums at the highest-paying firms (in the 10th decile of the distribution of estimated firmeffects). The largest mean residuals are for the lowest-ability workers in the lowest payingfirms – an effect which may reflect the impact of the minimum wage in Portugal. Residualplots for workers and firms in Germany, reported by CHK, and in Italy, reported by Macisand Schivardi (2016), also show no evidence of systematic departures from the predictionsof a simple AKM style model.

A different approach to assessing the additive separability assumption comes from Bon-homme, Lamadon, and Manresa (2015) who estimate a worker-firm model with discreteheterogeneity where each pairing of worker and firm type is allowed a different wage effect.Their results indicate that an additive model provides a very good approximation to Swedish

17

employer-employee data – allowing interactions between worker and firm type yields a triv-ial (0.8%) increase in explained wage variance. Lochner and Shulz (2016) reach a similarconclusion using information on relative wage rankings inside the firm and firm value addedto infer worker and firm types. They find using German data that additive separabilityprovides a good approximation to the wage structure except for the lowest skilled workers.

Though these results suggest that firm effects are, on average, similar for different typesof workers, there is of course scope for differences to emerge in selected subpopulations.For example, Goldschmidt and Schmieder (2015) find in large German firms that food,cleaning, security, and logistics (FCSL) workers exhibit different wage fixed effects than otheroccupations. Specifically, the firm wage effects of FCSL workers are attenuated relative tonon-FCSL workers. Likewise, Card, Cardoso, and Kline (2016) find that Portuguese womenexhibit slightly attenuated firm effects relative to men, which they argue reflects genderdifferences in bargaining behavior.

3 Reconciling rent-sharing estimates with results from

studies of firm switching

In their original study AKM showed that the estimated firm-specific wage premiums werepositively correlated with measures of firm profitability including value added per workerand sales per worker. A number of more recent studies have also confirmed that there is apositive link between firm-specific pay policies and productivity (e.g., Cahuc, Postel-Vinay,and Robin, 2006; Bagger, Christensen, and Mortensen, 2014).

To further bridge the gap between the rent-sharing literature and the firm-wage effectsliteratures we conducted a simple exercise using data on male workers in Portugal observedin the QP between 2005 and 2009 (i.e., the same data used in Panel A of Table 2). The AKMmodel posits that the log of the wage of a given worker in a given year can be decomposedinto the sum of a person effect, a firm or establishment effect, a time-varying index of personcharacteristics, and a residual that is orthogonal to the firm and person effects. It followsthat the rent sharing elasticity obtained from a regression of wages on a time-invariantmeasure of rents at the current employer (γw) can be decomposed into the sum of threecomponents reflecting the regression on firm-specific rents of the estimated worker effects(γα), the estimated firm effects (γψ), and the time-varying covariate index (γXβ):

γw = γα + γψ + γXβ.

The regression coefficients γα and γXβ represent sorting effects. To the extent that firms with

18

larger measured rents hire older workers, or workers with greater permanent skills, γα and/orγXβ will be positive. The coefficient γψ, on the other hand, is arguably a clean measure ofthe rent sharing elasticity, since ψJ(i,t) represents a firm-specific wage premium that is paidon top of any reward for individual-specific skills.

To implement this idea we use the estimated AKM parameters from Card, Cardoso andKline (2016), which were estimated on a sample that includes virtually all the observationsused for the cross-sectional models in Panel A of Table 2.15 The results are presented inPanel A of Table 4. Row 1 of the table reports estimated rent sharing elasticities using thelog hourly wage of each worker as a dependent variable. As in Table 2, we report threespecifications corresponding to models with only simple human capital controls (column1), controls for major industry and city (column 2) and controls for detailed industry andlocation (column 3). The estimated rent sharing elasticities in row 1 are qualitatively similarto the estimates in row 1 of Table 2 but differ slightly because the AKM model estimates arenot available for all workers/firms. Rows 2-4 show how the overall rent sharing elasticitiesin row 1 can be decomposed into a worker quality effect (row 2), a firm wage premium effect(row 3), and an experience-related sorting effect (row 4) which is close to 0.

A key conclusion from these estimates is that rent sharing elasticities estimated froma cross-sectional specification incorporate a sizable worker quality bias. In each columnof Table 4, roughly 40% of the overall wage elasticity in row 1 is due to the correlation ofworker quality (measured by the person effect component of wages) with firm specific quality.Adjusting for worker quality, the estimates in row 3 point to a rent sharing elasticity in therange of 0.10 to 0.15 – large enough to create a Lester range of wage variation of 16 to 24log points associated with the differences between firms at the 90th and 10th percentiles oflog value added per worker.

While the AKM approach reduces the estimated rent sharing elasticities substantially,the estimates in row 3 of Table 4 are still substantially larger than the within-job elasticitiesreported in Panel B of Table 2. There are several possible explanations for this gap. One isthat the within-job estimates are biased downward by measurement errors which comprise apotentially large share of the variance in relatively short-horizon changes in rents. A relatedexplanation, emphasized by Guiso, Pistaferri and Schivardi (2005) is that wages tend toadjust less to purely transitory fluctuations than to persistent changes in productivity.16

To the extent that industry-wide productivity shifts are more persistent than firm-specific15The sample used by Card, Cardoso, and Kline (2016) is slightly different than the sample of firms with

financial data we use in this paper, so the adding up constraint does not have to hold exactly. However, inall cases it holds approximately.

16Cardoso and Portela (2009) find evidence for this pattern using Portuguese worker firm data derivedfrom the QP.

19

within-industry shifts, this explanation can also account for the pattern of smaller elasticitieswhen more detailed industry controls are added to a rent-sharing model.

A third explanation is that some share of the firm-specific wage premium paid by moreproductive firms is a compensating differential for extra work effort or less desirable workingconditions (Rosen, 1986; Hwang et al., 1998). How much of the gap between firm stayerdesigns and worker switching designs such motives can explain remains an open question(Bonhomme and Jolivet, 2008; Lavetti and Schmutte, 2016; Sorkin, 2015; Taber and Vejlin,2016). In a recent field experiment, Mas and Pallais (2016) find that workers are not willingto pay much for scheduling flexibility but are willing to pay to avoid working evenings andweekends. In line with this evidence, Card, Cardoso, and Kline (2016, Appendix Table B6)examine the relationship between average hours of work and the estimated pay premiumsoffered by different firms in Portugal and find no evidence of compensating differentials forlong hours. Nevertheless, we cannot rule out some role for compensating differentials, whichwould imply that the estimates in row 3 of Table 4 may overstate the true rent-sharingelasticity. What is clear is that jobs with higher wage premiums last significantly longer(Card, Heining and Kline 2012, Appendix Table 8) which we take as providing relativelystrong evidence that the AKM firm effect estimates capture a significant rent component.

Differential rent sharing

We can use the AKM framework to examine another interesting question: to what extentdo different groups of workers receive larger or smaller shares of the rents at different firms?To do this, we fit separate AKM models for less-educated men (with less than a high schooleducation) and more-educated men (with high school or more) to our Portuguese wagesample. We then re-estimated the same rent sharing specifications reported in Panel A ofTable 4, separately for the two groups. The results are reported in Panels B and C of Table4.

The estimates reveal several interesting patterns. Most importantly, although the cor-relation between wages and value added per worker is a little higher for the more educatedmen, virtually all of this gap is due to a stronger correlation between the worker qualitycomponent of wages and value added. The correlations with the firm-specific pay premiumsare very similar for the two education groups. Thus, we see no evidence of differential rentsharing.

This finding is illustrated in Figure 8, which shows a binned scatterplot of mean logvalue added per worker at different firms (on the horizontal axis) versus the relative wagepremium for high-educated versus low-educated men at these firms. We also super-impose a

20

bin-scatter of the relative share of high educated workers at different firms (including bothmen and women in the employment counts for the two education groups). The relative wagepremium is virtually flat, consistent with the regression coefficients in rows 7 and 11 of Table4, which show nearly the same effect of value added per worker on the wage premiums for thetwo education groups. In contrast, the relative share of highly educated workers is increasingwith value added per worker – a pattern we interpret as largely driven by the “labor quality”component in value added per worker.17

4 Imperfectly competitive labor markets and inequality

With this background in mind we now turn to the task of developing a simple modelingframework that is useful for organizing and interpreting the empirical literature on firm-specific productivity and wage dispersion. In contrast to much of the rent sharing literature,we assume that wages are set by employers to maximize profits, subject to constraints onthe relationship between wages and the supply of labor. Rather than build a model of thesupply side based on search frictions, we follow the Industrial Organization literature byworking with a static “differentiated products” model that focuses on heterogeneity acrossworkers in their valuation of jobs at different employers. This differentiation endows firmswith some power to set wages as in classic monopsony models.18

While empirical work on monopsony has experienced something of a renaissance (seeManning, 2011 for a review) there has, to our knowledge, been little attempt to use thesemodels to reconcile facts in the literature on matched employer-employee data. We showthat static monopsony models can generate empirically plausible connections between firmproductivity and wages - observationally equivalent to simple rent-sharing models such asequation (1). They also, under reasonable assumptions, generate the prediction that wagesare additively separable in worker and firm heterogeneity, at least within broad skill groups.

A limitation of our framework relative to modern wage posting models is that we assume17As discussed in Section 1, value added per worker will in general depend on the “quality” of the workforce.

For example, if V Aj = PjTj [(1 − θ)Lj + θHj ] where Lj and Hj are the numbers of low and high skilledworkers employed at the firm, and Nj = Lj + HJ , then ln(TFPj/Nj) = ln(TFPj) + ln(qj) where qj =((1 − θ)Lj + θHj)/Nj is a measure of the quality of the workforce at firm j. The expected slope of aregression of the log of the relative share of highly educated workers on the log of value added per worker istherefore positive, even if there is no correlation between TFP and the share of highly educated workers.

18In this respect, our approach is akin to the classic Albrecht-Axell (1984) model of wage posting withleisure heterogeneity. However, because we allow for continuous heterogeneity in worker preferences, firmsare not indifferent between wage strategies and will mark wages down below marginal product accordingto the usual monopsonistic pricing rule. Our assumption that firms are ignorant about worker reservationvalues lies in contrast to the model of Postel-Vinay and Robin (2002) who assume that firms observe aworker’s outside option and offer wages that make them indifferent about accepting jobs.

21

all between-firm heterogeneity arises from heterogeneity in TFP or differences in the elastic-ity of labor supply to the firm. While this allows us to focus on the links between dispersionin productivity and wages it is important to remember that firms may also exhibit dispersionin wage policies for reasons having nothing to do with their production technology. Indeed,in the simplest version of Burdett and Mortensen’s (1998) model, firms are homogenous andthe identity of high wage and low wage firms is arbitrary.19

Market structure

There are J firms and two types of workers: lower-skilled (L) and higher-skilled (H). Eachfirm j ∈ {1, ..., J} posts a pair (wLj, wHj) of skill-specific wages that all workers costlesslyobserve. Hence, in contrast to search models, workers are fully informed about job oppor-tunities. As in many search models, however, we assume that firms will hire any worker (ofappropriate quality) who is willing to accept a job at the posted wage.

Firms exhibit differentiated work environments over which workers have heterogeneouspreferences. For worker i in skill group S ∈ {L,H}, the indirect utility of working at firm j

is:uiSj = βS ln(wSj − bS) + aSj + εiSj,

where bS is a skill-group specific reference wage level (arising for example from wages paid inan outside competitive sector) aSj is a firm-specific amenity common to all workers in groupS and εiSj captures idiosyncratic preferences for working at firm j, arising for example fromnon-pecuniary match factors such as distance to work or interactions with coworkers andsupervisors. We assume that the {εiSj} are independent draws from a type I Extreme Valuedistribution.20 Given posted wages, workers are free to work at any firm they wish. Hence,by standard arguments (McFadden, 1973), workers have “logit” choice probabilities of theform:

pSj ≡ P (argmaxk∈{1,..,J}

{uiSk} = j) =exp(βS(ln(wSj − bS) + aSj)∑Jk=1 exp(βS ln(wSk − bS) + aSk)

.

To simplify the analysis and abstract from strategic interactions in wage-setting, weassume that the number of firms J is very large, in which case the logit probabilities are

19We have also ignored “efficiency wage” explanations for firm wage premia which can emerge, for example,due to monitoring problems. See Akerlof and Yellen (1986) and Katz (1986) for reviews and Piyapromdee(2013) for an attempt to combine efficiency wage mechanisms with wage posting models.

20A potentially important distinction across worker subgroups is the relative magnitude of their variationin idiosyncratic preferences. Assume for example that uiSj = β0

S ln(wSj−bS)+aSj+τSεiSj , where τS is largerfor groups with a wider variation in idiosyncratic preferences, and εiSj is again Extreme Value distributed.This gives rise to the same choice probabilities as preferences uiSj = (β0

S/τS) ln(wSj− bS)+(1/τS)aSj+ εiSj .

22

closely approximated by exponential probabilities:

pSj ≈ λS exp(βS ln(wSj − bS) + aSj),

where (λH , λL) are constants common to all firms in the market. Thus, for large J , theapproximate firm-specific supply functions are:

lnLj(wLj) = ln(LλL) + βL ln(wLj − bL) + aLj (5)

lnHj(wHj) = ln(HλH) + βH ln(wHj − bH) + aHj, (6)

where L and H give the total numbers of lower-skilled and higher-skilled workers in themarket.21 Note that as βL, βH →∞ these supply functions become perfectly elastic and weapproach a competitive labor market with exogenous wages bL and bH .

Firm optimization

Firms have production functions of the form:

Yj = Tjf(Lj, Hj), (7)

where Tj is a firm-specific productivity shifter. We assume that f (., .) is twice differentiableand exhibits constant returns to scale with respect to Lj and Hj. For simplicity we alsoignore capital and intermediate inputs.22

The firm’s problem is to post a pair of skill-specific wages that minimize the cost of laborservices given knowledge of the supply functions (5) and (6). Firms cannot observe workers’preference shocks {εiSj}, which prevents them from perfectly price discriminating againstworkers according to their idiosyncratic reservation values. The firm’s optimal wage choices

21Berry and Pakes (2007) contrast demand models where consumers have idiosyncratic preferences forspecific products with what they term the “pure characteristics” model where consumers only care about afinite set of product characteristics. In the latter case, as the number of products grows large the demandelasticity tends to infinity – a phenomenon discussed in the labor market setting by Boal and Ransom (1997).We suspect the pure characteristics model is less applicable to the worker’s choice of employer because of themany non-pecuniary aspects of work that can give rise to match effects. For example, no two employers haveexactly the same location and workplace culture. However, it is clearly an important question for futureresearch which model works better empirically.

22This specification is appropriate if the user cost of capital and the prices of intermediate inputs are fixedand the firm’s output is a Cobb-Douglas function of these factors and the labor aggregate Tjf(Lj , Hj). Inthis case capital and intermediate inputs will adjust proportionally to Tjf(Lj , Hj).

23

solve the cost-minimization problem:

minwLj ,wHj

wLjLj(wLj) + wHjHj(wHj) s.t. Tjf(Lj(wLj), Hj(wHj)) ≥ Y.

The associated first order conditions can be written:

wLj1 + eLjeLj

= TjfLµj (8)

wHj1 + eHjeHj

= TjfHµj (9)

where eLj and eHj represent the elasticities of supply of L and H workers at the optimalchoice of wages, and µj represents the marginal cost of production, which the firm will equateto marginal revenue at an optimal choice for Y . Thus the terms TjfLµj and TjfHµj on theright hand sides of equations (8) and (9) represent the marginal revenue products of the twotypes of labor, while the terms on the left hand sides represent their marginal factor costs.Using equations (5) and (6), the elasticities of supply are:

eLj =βLwLjwLj − bL

eHj =βHwHjwHj − bH

.

Note that, for both groups, labor supply to the firm becomes infinitely elastic as wagesapproach the reference wage level bS. Using these expressions, the firm’s first order conditionscan be re-written:

wLj =1

1 + βLbL +

βL1 + βL

TjfLµj (10)

wHj =1

1 + βHbH +

βH1 + βH

TjfHµj. (11)

The firm’s optimal wage choice for skill group S is a weighted average of the reference wagebS and the group’s marginal revenue product. As βS →∞, eSj →∞ and the firm pays thereference wage bS. In the case where the reference wage is 0, the labor supply function forgroup S has a constant elasticity βS and the first order condition sets the wage equal to aconstant fraction βS/(1 + βS) of the marginal revenue product.

Note that firms post wages with knowledge of the shape of the skill-specific supply sched-ules but not the identities of the workers who comprise them. The last worker hired isindifferent about taking the job but the other employees strictly prefer their job to outside

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alternatives. These inframarginal workers capture rents by means of an information asym-metry: they hide from their employer the fact that they would be willing to work for a lowerwage.

We now derive results under various specifications of the production function and thefirm’s marginal revenue function. On the technology side, we start with a simple baselinecase where f (., .) is linear in Lj and Hj. This corresponds to a standard efficiency unitsmodel of the labor market in which lower and higher-skilled workers are perfect substitutes.We then consider the more general case where f (., .) is a CES production function. On therevenue side, we initially assume that the firm faces a fixed output price. We then considerthe more general case where the faces a constant elasticity product demand function.

Baseline case: linear production function and fixed output price

To develop intuition, we begin with the simplest possible example where the firm faces afixed output price P 0

j and has a linear production function:

Yj = TjNj = Tj((1− θ)Lj + θHj).

Here Nj represents the efficiency units of labor at the firm and the parameter θ ∈ (0.5, 1),which we assume is common to all firms, governs the relative productivity of the two types oflabor. Under this specification of technology and market structure, the first order conditions(10) and (11) evaluate to:

wLj =1

1 + βLbL +

βL1 + βL

TjP0j (1− θ)

wHj =1

1 + βHbH +

βH1 + βH

TjP0j θ.

The determination of the optimal wage in the simplified situation where there is only oneskill group is illustrated in Figure 9. The firm faces an upward sloping inverse labor supplyfunction of the form w = b + N1/β. The associated marginal factor cost is MFC = b +1+ββN1/β. The firm equates MFC with marginal revenue product (MRP ) leading to an

equilibrium wage w = 11+β

b + β1+β

MRP . As shown in the Figure, if the firm’s marginalrevenue product increases, both employment and wages will increase at the firm. In contrastto traditional rent-sharing models, however, this positive relationship between wages andproductivity does not stem from wage bargaining. Firms unilaterally post profit-maximizingwages that leave the marginal worker with no surplus on the job. The firm shares rents withinframarginal workers only because it lacks the information necessary to price discriminate

25

based upon reservation wages.To understand the implications of this model for the relative wage structure, suppose that

the reference wages of the two skill groups are proportional to their relative productivities,so that

bL = (1− θ)b, bH = θb.

This restriction is natural if one views bS as an outside wage that can be earned in a fullycompetitive sector where wages equal marginal products. Now the first order conditions canbe re-written:

lnwLj = ln(1− θ)b1 + βL

+ ln(1 + βLRj) (12)

lnwHj = lnθb

1 + βH+ ln(1 + βHRj) (13)

where Rj ≡TjP

0j

bgives the proportional gap in marginal labor productivity at firm j relative

to the competitive sector. Wages of both skill groups contain a rent-sharing component thatdepends on Rj and the skill group-specific supply parameter βS.

Note that under the linear technology assumption, value added per standardized unit oflabor is vj ≡ P 0

j Yj/N j = P 0j Tj, so Rj = vj/b is the ratio of value added per standardized

unit of labor to the outside wage for a worker with 1 efficiency unit of labor. Equations (12)and (13) therefore imply that the elasticity of wages of skill group S with respect to valueadded per worker is:

ξSj ≡∂ lnwSj∂ ln vj

=βSRj

1 + βSRj

.

Interestingly, this is the same as the expression for the rent-sharing elasticity (equation 2) ina bargaining model where workers are assumed to capture a fixed share of the quasi-rents.

The estimated rent sharing elasticities in Table 1 (and in our reanalysis of Portuguesedata) indicate that a typical value of this elasticity is around 0.10, which suggests that theaverage value of βSRj is also around 0.10.23 Hence, an average worker earns about 10%higher wages than he or she would earn at the lowest-wage firms in the economy that havevj = b. This estimate of average rents earned per worker is remarkably consistent with theestimates of Card, Cardoso and Kline (2016, Table III) obtained by benchmarking wagepremiums at firms in the Portuguese economy relative to the premiums paid by the leastprofitable firms in the country. By contrast Hornstein, Krussell, and Violante (2011) show

23This is an approximation, as the model implies rent sharing elasticities vary systematically across firmsin relationship to productivity. Supposing for simplicity that an average elasticity is being reported inempirical studies, the approximation that 1

J

∑j βSRj ≈

1J

∑j ξSj will be accurate when the distribution of

ξSj is bounded from above by a number far below one.

26

that a wide class of search models have difficulty generating mean wages more than 5% abovethe lowest offered wage.

The elasticity of labor supply for skill group S when wages are determined by the firstorder conditions (12) and (13) is:

eSj =βLwSjwSj − bS

=1 + βSRj

Rj − 1.

Assuming that βSRj = 0.1, a value of the firm-specific elasticity of supply of around 4 impliesthat Rj ≈ 1.3 and βS ≈ 0.08. While many empirical estimates of the elasticity of supply tothe firm are lower than 4 (Manning, 2011), we consider this a reasonable “near competitive”benchmark as it implies an equilibrium markdown of wages relative to marginal products ofonly 20%.

A key implication of equations (12) and (13) is that when βL = βH , the relative wages ofthe two skill groups are independent of firm-specific productivity. To simplify the discussion,assume that βLRj and βHRj are both relatively small (i.e., on the order of 0.10). In such acase the Taylor approximation:

lnwLj ≈ ln(1− θ)b1 + βL

+ βLRj

lnwHj ≈ lnθb

1 + βH+ βHRj

will be highly accurate. This implies the log wage gap between high and low skilled workersat firm j is:

lnwHjwLj

= lnθ

1− θ+ ln

1 + βL1 + βH

+ (βH − βL)Rj. (14)

When βL = βH = β, wages can be written in the form:

lnwSj = αS + ψj, (15)

where αS ≡ ln b1+β

+1[S = L]× ln(1− θ) + 1[S = H]× ln θ is a skill-group-specific constant,and ψj = βRj =

βbvj is the firm-specific wage premium paid by firm j. This simple model

therefore yields a reduced form specification for individual wages that is consistent with theadditively separable formulation proposed by AKM. Moreover, the firm effects should bestrongly related to value added per worker, something we saw evidence for in Table 4.24

When one group has a higher value of the supply parameter β, the log wage gap between24While Table 4 regresses the firm effects on ln vj , the above analysis suggests ψj should be nearly linear

in vj with a coefficient of approximately β/b. In practice, Card, Cardoso, and Kline (2016) find that AKMfirm effects are instead approximately linear in ln vj above a minimal threshold level.

27

workers in different skill groups will be higher at more profitable firms. In this case, the datawill be described by an AKM-style model with skill-group-specific firm effects. The wagepremium for skill group S at firm j will be:

ψSj = βSRj.

The value of the ratio βH/βL can be identified from a projection of the firm effects forworkers in group H on the associated firm effects for workers in group L (since ψHj = βH

βLψLj ).

Card, Cardoso and Kline (2016, Figure V) relate gender-specific firm effect estimates to oneanother and find results consistent with a value of β about 10% larger for males than females.

Between-Firm Sorting

Even when βL = βH and the wage gap between workers in the two skill groups is constant atany given firm, themarket-wide average wage for each skill group will depend on their relativedistribution across firms. In particular, (15) implies the expected log wage for workers inskill group S is:

E[lnwSi] = αS +∑j

ψjπSj

where πSj is the share of workers in skill group S employed at firm j. Thus the market-widewage differential between high and low skilled workers depends on their relative productivity,their relative supply elasticities, and the relative shares of the two groups employed at firmswith higher or lower wage premiums:

E[lnwHi]− E[lnwLi] = αH − αL +∑j

ψj(πHj − πLj). (16)

The third term in this expression represents a between-firm sorting component of the averagewage gap. Card, Cardoso, and Kline (2016) show that 15-20% of the wage differentialbetween men and women in Portugal is explained by the fact that males are more likelyto work at firms that pay higher wage premiums to both gender groups. Similarly, Card,Heining and Kline (2012) show that an important share of the rising return to education inGermany is explained by the increasing likelihood that higher-educated workers are sortedto establishments with higher pay premiums.

Some simple evidence on the importance of the sorting component for the structure ofwages for Portuguese male workers is presented in Figure 10. Here, we plot the mean firmeffects by age for Portuguese men in 5 different education groups. We normalize the estimated

28

firm effects using the procedure described in Card, Cardoso and Kline (2016), which setsthe average firm effect to zero for firms in (roughly) the bottom 15% of the distribution oflog value added per worker. The figure shows two important features. First, within eacheducation group, the mean firm effect associated with the jobs held by workers at differentages is increasing until about age 50, then slightly decreasing.25 Thus, the lifecycle patternof between-firm sorting contributes to the well-known shape of the lifecycle wage profile.Second, at all ages, more highly educated workers are more likely to work at firms that payhigher wage premiums to all their workers. A significant share of the wage gap between menwith different education levels is therefore attributable to differential sorting.

When the supply parameter β varies across groups the wage decomposition will containan additional term, reflecting a weighted average across firms of the rent sharing componentsof the two skill groups:

E[lnwHi]− E[lnwLi] = αH − αL +∑j

ψLj (πHj − πLj) +∑j

(ψHj − ψLj )πHj

= αH − αL +∑j

ψHj (πHj − πLj) +∑j

(ψHj − ψLj )πLj.

As in a traditional Oaxaca (1973)-style decomposition these expressions give alternative waysto evaluate the contributions of differences in the distributions of the two skill groups acrossfirms, and differences in the return to working at a given firm between the two groups.

Downward-sloping firm-specific product demand

So far we have assumed that the firm is a price taker in its output market. Suppose now thatthe firm faces an inverse demand function Pj = P 0

j Yj−1/ε with ε > 1 giving the elasticity of

product demand. This yields the marginal revenue function:

MRj =

(ε− 1

ε

)P 0j Y

−1/εj

25Topel and Ward (1992) showed that job-to-job mobility was an important component of wage growthfor young men in the U.S. labor market. They interpreted their finding as mainly arising from gains in thejob-match component of wages, rather than as systematic mobility to firms that pay higher wages to allworkers.

29

In this case, assuming as above that bL = (1 − θ)b and bH = θb, the first order conditions(10) and (11) evaluate to:

wLj =b(1− θ)1 + βL

(1 + βL

(ε− 1

ε

)TjP

0j Y

−1/εj ) (17)

wHj =bθ

1 + βH(1 + βH

(ε− 1

ε

)TjP

0j Y

−1/εj ). (18)

These equations can be simplified by noting that value-added per efficiency unit of labor is:

vj ≡PjYjNj

= P 0j Tj Yj

−1/ε.

Thus, the optimal choices for wages can be written:

lnwLj = ln(1− θ)b1 + βL

+ ln(1 + βLR′j)

lnwHj = lnθb

1 + βH+ ln(1 + βHR

′j).

where R′j = ( ε−1ε)vj/b. Note that as ε → ∞, these reduce to equations (12) and (13).

Morever, regardless of the value of ε, if βL ≈ βH , then relative wages are constant acrossfirms and the AKM model of the wage structure remains valid with the firm effects beingmonotone functions of value added per worker.

The implied elasticity of wages of skill group S with respect to value added per standard-ized unit of labor is:

ξSj =∂ lnwSj∂ ln vj

=βSR

′j

1 + βSR′j.

Assuming this elasticity is approximately 0.10 suggests that βSR′j ≈ 0.10. Moreover, theelasticity of labor supply of skill group S to the firm is:

eSj =βLwSjwSj − bS

=1 + βSR

′j

R′j − 1,

so calibrating this elasticity to a value of 4 would suggest that R′j = 1.28, again pointing toa value of βS ≈ 0.08. Finally, note that the elasticity of employment of skill group S withrespect to a change in vj is

eSjξSj =βSR

′j

R′j − 1

which has a value of approximately 4 under the preceding assumptions.When the firm faces a downward-sloping product demand, value added per efficiency unit

30

of labor (vj) depends on the endogenous choice of output. In the Appendix, we show thatthe elasticities of vj with respect to an exogenous shift in output demand (indexed by P 0

j )

or an exogenous increase in productivity (indexed by Tj) are:

∂ ln vj∂ lnP 0

j

=ε

ε+mj

∂ ln vj∂ lnTj

=ε− 1

ε+mj

wheremj ≡ ∂ lnNj

∂ ln vj=

R′j

R′j−1

[βL(1−θ)(LjNj )+βHθHjNj]measures the rate at which overall efficiency

units of labor expand when there is an exogenously driven increase in value added. Fromthese expressions it follows that the elasticities of the wages of skill group S with respect todemand shocks and productivity shocks are:

∂ lnwSj∂ lnP 0

j

=ε

ε+mj

× ξSj

∂ lnwSj∂ lnTj

=ε− 1

ε+mj

× ξSj

Under the calibrations above, mj is approximately 4. Assuming the firm-specific productdemand elasticity is between 3 and 10 the elasticity of wages with respect to a shift in thefirm’s demand curve will be between 0.04 and 0.07, and the elasticity with respect to a shiftin technological efficiency will be between 0.035 and 0.065.

Imperfect substitution between skill groups

A limitation of our baseline model is that it assumes perfect substitutability between thetwo skill groups. We now extend the model by assuming that the firm’s output is a CESaggregate of high and low skilled labor:

Yj = TjNj = Tjf(Lj, Hj)

f(Lj, Hj) = [(1− θ)Lρj + θHρj ]

1/ρ (19)

where ρ ∈ (−∞, 1] and σ = (1 − ρ)−1 is the elasticity of substitution between the types oflabor. The marginal productivities of the two groups take the form:

TjfL = Tj(1− θ)Lρ−1j Nj1−ρ

TjfH = TjθHjρ−1Nj

1−ρ .

31

Assuming that the firm faces a constant price P 0j for its output, and that bL = (1− θ)b and

bH = θb, the first order conditions (10) and (11) evaluate to:

wLj =b(1− θ)1 + βL

(1 + βLRj (Lj/Nj)− 1σ ) (20)

wHj =bθ

1 + βH(1 + βHRj (Hj/Nj)

− 1σ ) (21)

where Rj = TjP0j /b = YjP

0j /bNj = vj/b is value added per standardized unit of labor relative

to the reference wage. These differ from the corresponding equations with a linear technology(equations (12) and (13)) by the terms (Lj/Nj)

− 1σ and (Hj/Nj)

− 1σ which adjust the marginal

productivities of L and H workers based upon their relative employment shares. These termsdisappear when Lj = Hj, or when σ is large.26

Note that even when βL = βH , log wages in this model are non-separable in workerand firm heterogeneity. This is because firm heterogeneity in factor proportions leads tofirm specific wage-skill gaps of the sort usually analyzed at the market level (e.g., Katz andMurphy, 1992). If skill types were observable, it would be natural to estimate such a modelvia nonlinear least squares using data on firm value added. With unobserved skill types, aninteractive fixed effects specification would be required that allows the firm effects to dependon the unobserved skill ratio at the firm.

To derive the rent sharing elasticities in this model, define:

τLj =βLRj (Lj/Nj)

− 1σ

1 + βLRj (Lj/Nj)− 1σ

τHj =βHRj (Hj/Nj)

− 1σ

1 + βHRj (Hj/Nj)− 1σ

These are the elasticities of wages with respect to vj, ignoring any adjustment to the relativeinput of L and H labor. They also represent the proportional wage premiums for L and Hworkers associated with working at a firm with R = Rj relative to a marginal firm with Rclose to 1. With this notation, we show in the Appendix that the elasticities of wages withrespect to value added per labor input can be expressed as:

ξLj ≡∂ lnwLj∂ ln vj

=τLj(1 +

τHjeHjσ

)

1 + 1σ((1− κj)τLjeLj + κjτHjeHj)

ξHj ≡∂ lnwHj∂ ln vj

=τHj(1 +

τLjeLjσ

)

1 + 1σ((1− κj)τLjeLj + κjτHjeHj)

,

26Defining hj = Hj/Lj , note that Lj/Nj = (1− θ(1−hρj ))−1/ρ and Hj/Nj = hj(1− θ(1−hρj ))−1/ρ. Thus,when hj = 1, Lj/Nj = Hj/Nj = 1.

32

where (as above) eLj and eHj are the elasticities of labor supply of L and H workers to thefirm, and κj ≡

(1−θ)Lρj(1−θ)Lρj+θH

ρj= ∂ ln f

∂ lnLj= 1− ∂ ln f

∂ lnHj. Notice that:

limσ→∞

ξSj =βSRj

1 + βSRj

which is the expression derived above for our baseline case with a linear technology. Withimperfect substitution between groups the value-added elasticities of the two skill groups,ξLj and ξHj , will depend on τLj and τHj and on the labor supply elasticities of the twogroups.

The general effect of allowing imperfect substitution between L andH labor is to generatevalue added elasticities that are smaller than the “rent shares” τLj and τHj, but of similarmagnitudes. For example, consider a calibration with θ = 0.6, σ = 1.4, and an averageworkforce comprised of 60% L workers and 40%H workers. In this setup, a value of Rj = 1.45

combined with βL = 0.09 and βH = 0.15 yields τL = 0.10 and τH = 0.20, implying that lower-skilled workers earn a 10% wage premium relative to their outside wage, while higher-skilledworkers earn a 20% premium. The implied firm-specific labor supply elasticities of the twogroups are 5 and 2, respectively, while the implied elasticities of their wages with respect tovalue added per worker are 0.05 and 0.11, respectively.

To summarize: a CES technology will give rise to generalized AKM models in whichthe wage differentials between different skill groups vary across firms, and the elasticity ofwages with respect to value added per worker will also vary across groups and across firms,depending on the firm-specific skill employment shares at the firm. Nevertheless, the basicintuition of our benchmark model remains. In particular, a simple model of monopsonisticwage setting can explain the existence and quantitative magnitude of systematic firm wagepremiums that are highly correlated across skill groups and highly correlated with measuresof firm-specific productivity.

Relationship to other models and open questions

Although we have worked with a static model of employer differentiation, there are obviousbenefits to considering more realistic dynamic models, not least of which is that they providea rationale for the worker mobility typically used to estimate firm wage effects. Appendix Bconsiders a simple dynamic extension of our framework that yields random mobility betweenfirms and has essentially identical steady state implications for wages and employment.However, it would be interesting to consider richer models where workers systematicallyclimb a productivity job ladder and can spend some time unemployed. Another interesting

33

extension would be to allow incumbent workers to face switching costs that lead firms to pricediscriminate against them. This could lead to offer matching behavior as in Postel-Vinayand Robin (2002) and to new predictions about recruitment and retention policies.

By assuming the number of employers is very large, we have adopted a partial equilibriumframework with no strategic interactions between employers. With a finite number of firms,a shock to one firm’s productivity will affect the equilibrium employment and wages ofcompetitor firms. Staiger, Spetz, and Phibbs (2010) provide compelling evidence of suchresponses in the market for nurses. As in the oligopoly literature, analysis of a finite employermodel with strong strategic dependence may be complicated by the presence of multipleequilibria, which requires different methods for estimation (e.g., Ciliberto and Tamer, 2009)but may also yield interesting policy implications.

Finally, it is worth noting some links between our modeling of workplace differentia-tion with the literature on compensating differentials for non-wage amenities (Rosen, 1986;Hwang, Mortensen and Reed, 1998). In our model, non-wage amenities that are valuedequally by all workers simply shift the intercept of the labor supply curve to the firm. Buta monopsonist firm sets wages based upon the elasticity of labor supply to the firm, whichis governed entirely by the distribution of taste heterogeneity. For this reason, our modelexhibits no compensating differentials of the standard sort. Amenities affect firm effectsonly through their influence on TFP – a firm with attractive non-wage amenities will growlarge which should depress its revenue productivity and therefore lower its firm wage effect.Empirically distinguishing this effect, which is mediated through product prices, from thestandard compensation mechanism is policy relevant since the monopsony model will tendto imply a different incidence of (say) employer provided health benefits on workers than acompensating differentials model.

5 Conclusion

There is no doubt that a large share of wage inequality is driven by differences in worker skills.But economists have long had evidence (e.g., Lester, 1946; Slichter, 1950) that employercharacteristics exert an independent effect on wages. While the ability of firms to set wagesis disciplined by market competition, there are clearly limits to those competitive forces,which also fail to eliminate productivity and output price differences across firms (Hsieh andKlenow, 2009).

Modern search theory provides one rationale for the existence of wage setting power(Mortensen, 2005). But even without search frictions, firms will be able to set wages if (asseems likely) workers differ in their valuation of firms’ non-wage characteristics. While the

34

mechanisms giving rise to market power under these two approaches are different, both implythat labor is supplied inelastically to firms, providing some scope to set wages. As noted byManning (2011) such market power can generate a positive relationship between firm-specificproductivity and wages that mimics models in which workers bargain with employers overwages. But in our setting firms set wages unilaterally and rents are shared only because ofinformation asymmetries. We believe that this alternative explanation for what the literaturehas called “rent sharing” is more plausible than one based on worker bargaining power,particularly in economies that lack strong unions and in settings (like Portugal) where mostfirms pay wages above those required by sectoral bargaining agreements. Indeed, structuralestimates of worker bargaining strength are often quite low (e.g., Cahuc, Postel-Vinay, andRobin, 2006). One interesting implication of a monopsony-based explanation for the linkbetween productivity and wages is that there is no “hold-up” problem in the firm’s investmentdecision – a prediction consistent with the empirical findings of Card, Devicienti, and Maida(2014).

The empirical literature on firm wage inequality has progressed dramatically with theintroduction of huge matched employer employee datasets. Yet significant challenges remain.The field continues to rely almost exclusively on observational studies predicated on plausible,but ultimately debatable, identifying assumptions. More research is needed using researchdesigns that can credibly identify the causal link from firm-specific shocks to workers’ wages.Another outstanding goal is the development of studies that directly manipulate incentivesfor workers to leave and join particular firms, as in the innovative experimental design ofDal Bó, Finan, and Rossi (2013). Such designs can be used to rigorously assess the degreeof bias in observational firm switching designs.

While research on labor market inequality typically strives for general explanations ofnational trends, the way forward in this literature may not involve a “theory of everything”but rather more attention to the institutional details of particular labor markets. This is thetradition in the Industrial Organization literature, where studies generally focus on particularindustries, rather than the economy as a whole. It is plausible that firms have more wagesetting power in some labor markets than others and that the nature of firm wage vs. nonwagecompetition differs as well. A key empirical problem is how to define the labor market ofinterest, both geographically and with respect to skills. In the immigration literature, forexample, there is debate over whether labor markets are effectively national or local, andover how to classify different age, education, and national origin groups.27 Manning and

27For example, Borjas (2003) argues for national labor markets categorized into 5 education classes, withno distinction between immigrants and natives, whereas Ottaviano and Peri (2013) and Card (2009) arguefor city-level markets categoried into 2 education classes but stratified by immigrant origin.

35

Petrolongo (2011) develop a spatial job search model where markets can be geographicallyoverlapping. Fitting their model to spatially detailed data on job applications and vacancies,they find that workers are discouraged from searching in areas with strong competition fromother job seekers and that shocks to local neighborhoods can yield important “ripple effects”on labor market activity in nearby areas.28 New case studies of settings where the marketstructure of labor demand can be carefully documented would be particularly useful.29

Finally, the idea that even highly advanced labor markets like that of the United Statesmight be better characterized as imperfectly competitive opens a host of questions aboutthe welfare implications of industrial policies and labor market institutions such as theminimum wage, unemployment insurance, and employment protection (Katz and Summers,1989; Acemoglu, 2001; Coles and Mortensen, 2016). Empirical work lags particularly farbehind the theory in this domain. Additional evidence on how actual labor market policiesaffect firm and worker behavior is needed to assess the plausibility of these theoretical policyarguments.

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43

Appendix

A. Derivations

A.1 Downward-Sloping Product Demand Let κj =∂ lnNj

∂ lnLj= (1 − θ)Lj/Nj represent

the elasticity of total labor efficiency units with respect to low skilled labor, and notice thatthe elasticity of labor inputs with respect to high skilled labor is ∂ lnNj

∂ lnHj= θHj/Nj = 1− κj.

Suppose that the elasticities of wages of the two groups with respect to value added perworker are ξLj and ξHj, respectively, and that eL and eH are the elasticities of labor supplyof the two groups to the firm. Since the firm’s labor input choices are constrained by thelabor supply functions of L and H labor, the elasticity of total labor input with respect ashift in value-added per worker is:

mj ≡∂ lnN j

∂ ln vj=

∂ lnN j

∂ lnLj× ∂ lnLj∂ lnwLj

× ∂ lnwLj∂ ln vj

+∂ lnN j

∂ lnHj

× ∂ lnHj

∂ lnwHj× ∂ lnwHj

∂ ln vj= κjeLjξLj + (1− κj)eHjξHj.

Now, using the fact that value added per worker is vj = P 0j T

1−1/εj N j

−1/ε, it follows that:

∂ ln vj∂ lnP 0

j

= 1− 1

εmj

∂ ln vj∂ lnP 0

j

⇒ ∂ ln vj∂ lnP 0

j

=ε

ε+mj

and similarly:∂ ln vj∂ lnTj

=ε− 1

ε+mj

.

A.2 CES Technology

We now extend the model by assuming that the firm’s production f is in the CES class, sothe labor input at firm j is:

Nj = f(Lj, Hj) = [(1− θ)Lρj + θHρj ]

1/ρ

As noted in the text, the marginal products of the two skill groups are:

fL = (1− θ)Lρ−1j f(Lj, Hj)1−ρ

fH = θHjρ−1f(Lj, Hj)

1−ρ

44

Finally, define

κj ≡∂ lnNj

∂ lnLj=

(1− θ)Lρj(1− θ)Lρj + θHρ

j

and note that ∂ lnNj∂ lnHj

= 1-κj.The first order conditions (20) and (21) can be written as:

lnwLj = lnb(1− θ)1 + βL

+ ln(1 + βLRj (Lj/Nj)− 1σ )

lnwHj = lnbθ

1 + βH+ ln(1 + βHRj (Hj/Nj)

− 1σ )

Differentiating these equations and simplifying notation we obtain:[1 + 1

σ τL(1− κj)eL − 1σ τL(1− κj)eH

− 1σ τHκjeL 1 + 1

σ τHκjeH

][d lnwLj

d lnwHj

]=

[τL

τH

]dvj

Some manipulation establishes that:

∂ lnwLj∂ ln vj

=τLj(1 +

τHjeHjσ

)

1 + 1σ((1− κj)τLjeLj + κjτHjeHj)

∂ lnwHj∂ ln vj

=τHj(1 +

τLjeLjσ

)

1 + 1σ((1− κj)τLjeLj + κjτHjeHj)

.

B. Two Period Model of Supply

Here we consider a two-period extension of our static framework. A worker i of type S facesindirect utility over firms j ∈ {1, ..., J} of:

uiSj = βS ln (wSj − bS) + aSj + εiSj,

where εiSj is drawn from a type I Extreme Value distribution. Hence the period 1 choiceprobabilities are:

p1Sj =exp(βS ln

(w1Sj − bS

)+ a1Sj)∑J

k=1 exp(βS ln (w1Sk − bS) + a1Sk)

.

≈ λ1S exp(βS ln(w1Sj − bS

)+ a1Sj)

In the second period, a fraction π̃ of the workers get a new draw ε′i of idiosyncratic

45

Extreme Value preferences. Because each firm’s market share is very low, workers will onlychoose employers for which they have a very strong idiosyncratic taste. Hence, the chancesof preferring to stay at the same firm with a new taste draw are essentially zero. With thisin mind, we write second period market shares as:

p2Sj = π̃exp(βS ln

(w2Sj − bS

)+ a2Sj)∑J

l=1 exp(βS ln (w2Sl − bS) + a2Sl)

+ (1− π̃) p1Sj

≈ π̃λ2S exp(βS ln(w2Sj − bS

)+ a2Sj) + (1− π̃) p1Sj

Clearly as π̃ → 1, the labor supply function becomes static. Otherwise, we have a partialadjustment process that yields heterogeneity in the labor supply elasticity depending uponhow far p1Sj is from λ2S exp(βS ln (w

2Sk − bS) + a2Sk). In a steady state these two objects will

be the same and the elasticity of supply to each firm simplifies to π̃ times the usual staticelasticity eSj.

Therefore, we can think about the steady state of a dynamic model with taste shocks asone where firms face a supply curve eSjπ̃ and set wages accordingly. As before, firms cannotobserve workers’ preferences. Hence, employee threats to leave in response to taste shockswill not be viewed as credible by the firm despite the firm’s knowledge that a fraction π̃ ofworkers did in fact draw new tastes. Because the firm cannot budge in its wage policy, eachperiod will yield a fraction π̃ of workers switching between firms.

46

Figure 1: Trends in Between‐Establishment Dispersion in Wages and Productivity

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1977 1982 1987 1992 1997 2002 2007

Standard Deviatio

n Log Wages

0.50

0.55

0.60

0.65

0.70

0.75

0.80

Standard Deviatio

n Log Re

v/Worker

Std Dev. of Weekly Wage (left axis)

Std. Dev. of Revenue/Worker (right axis)

Source: Barth et al (2016)

Figure 2: Implied Age Profiles from AKM Models with Alternative Normalizations of the Age Profile (Men with Primary Education Only)

‐3.0

‐2.5

‐2.0

‐1.5

‐1.0

‐0.5

0.0

0.5

20 25 30 35 40 45 50 55 60

Age

Implied Ag

e Profile 

1980 birth cohort 1970 birth cohort 1960 birth cohort 1950 birth cohort

Blue squares = Baseline cubic  (age normalized around 40)

Yellow diamonds = Alternative cubic(unnormalized age)

Figure 3: Mean Log Wages of Portuguese Male Job Changers, Classified by Quartile of Co‐Worker Wages at Origin and Destination 

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

‐2 ‐1 0 1Time (0=first year on new job)

Mean Log Wage of M

overs

4 to 4

4 to 3

4 to 2

4 to 1

1 to 4

1 to 3

1 to 2

1 to 1

Notes: Figure shows mean wages of male workers at mixed‐gender firms who changed jobs in 2004‐2007 and held the preceding job for 2 or more years, and the new job for 2 or more years.   Job is classified into quartiles based on mean log wage of co‐workers of both genders. Source: Card, Cardoso and Kline (2016, Figure I).

Figure 4: Mean Wages of Portuguese Female Job Changers, Classified by Quartile of Co‐Worker Wages at Origin and Destination 

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

‐2 ‐1 0 1Time (0=first year on new job)

Mean Log Wage of M

overs

4 to 4

4 to 3

4 to 2

4 to 1

1 to 4

1 to 3

1 to 2

1 to 1

Notes: Figure shows mean wages of female workers at mixed gender firms who changed jobs in 2004‐2007 and held the preceding job for 2 or more years, and the new job for 2 or more years.   Jobs are classified into quartiles based on mean log wage of co‐workers of both genders. Source: Card, Cardoso and Kline (2016, Figure II).

Figure 5a: Test for Symmetry of Regression‐Adjusted Wage Changes of Portuguese Male Movers Across Coworker Wage Quartiles

‐0.5

‐0.4

‐0.3

‐0.2

‐0.1

0

0 0.1 0.2 0.3 0.4 0.5Mean Log Wage Change For Upward Movers

Mean Log Wage Ch

ange, D

ownw

ard Movers

Q1 to Q4,Q4 to Q1

Q2 to Q4,Q4 to Q2Q1 to Q3,

Q3 to Q1

Q3 to Q4,Q4 to Q3Q2 to Q3,

Q3 to Q2

Q1 to Q2,Q2 to Q1

Note: Figure plots regression adjusted mean wage changes over 4 year interval for job changers who move across coworker wage quartile groups indicated. Dashed line represents symmetric changes for upward and downward movers. Source: Card, Cardoso and Kline (2016, Appendix Figure B3).

Figure 5b: Test for Symmetry of Regression‐Adjusted Wage Changes of Portuguese Female Movers Across Coworker Wage Quartiles

‐0.4

‐0.3

‐0.2

‐0.1

0

0 0.1 0.2 0.3 0.4Mean Log Wage Change For Upward Movers

Mean Log Wage Ch

ange, D

ownw

ard Movers

Q1 to Q4,Q4 to Q1

Q2 to Q4,Q4 to Q2

Q1 to Q3,Q3 to Q1

Q3 to Q4,Q4 to Q3

Q2 to Q3,Q3 to Q2

Q1 to Q2,Q2 to Q1

Note: Figure plots regression adjusted mean wage changes over 4 year interval for job changers who move acrosscoworker wage quartile groups indicated. Dashed line represents symmetric changes for upward and downward movers. Source: Card, Cardoso and Kline (2016, Appendix Figure B4).

1 2 3 4 5 6 7 8 9 10

13

57

9

‐0.02

‐0.01

0

0.01

0.02

1 2 3 4 5 6 7 8 9 10

13

57

9

Firm Effect Decile

Person EffectDecile

Figure 6: Mean Residuals by Person/Firm Deciles,Portuguese Male Workers

Note: Figure shows mean residuals from estimated AKM model with cells defined by decile of estimated firm effects interacted with decile of estimated person effect. Source: Card, Cardoso and Kline (2016, Appendix Figure B5).

1 2 3 4 5 6 7 8 9 10

13

57

9

‐0.02

‐0.01

0

0.01

0.02

1 2 3 4 5 6 7 8 9 10

13

57

9

Firm Effect Decile

Person EffectDecile

Figure 7: Mean Residuals by Person/Firm Deciles,Portuguese Female Workers

Note: Figure shows mean residuals from estimated AKM model with cells defined by decile of estimated firm effects interacted with decile of estimated person effect. Source: Card, Cardoso and Kline (2016, Appendix Figure B6).

Figure 8: Relative Wage Premium and Relative Employment of High vs. Low Education Workers 

‐2.0

‐1.5

‐1.0

‐0.5

0.0

0.5

1.0

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6

Mean Log Value Added per Worker

Relativ

e Wage Prem

ium, Log

 Relative Em

ploymen

t Relative Wage Premium (High vs. Low Education Men)

Log Relative Employment (High vs. Low Education Males and Females)

Note: Firms are divided into 100 cells based on mean log value added per worker, 2005‐2009, with equal numbers of person‐year observations per cell.

Figure 9: Determination of Wages 

Employment

Wage,MRP,MFC

inversesupply

marginalfactor cost

w 0

L 0

b

MRP varieswith VA/worker

Figure 10: Mean Firm Effects by Age and Education Group, Portuguese Males

0.0

0.1

0.2

0.3

0.4

19 23 27 31 35 39 43 47 51 55 59 63

Age

Mean Firm

 Effe

ct (Log

 Wage Scale)

UniversityCompleted High SchoolIntermediate Schooling (9 yrs)Upper Primary (6 yrs)Lower Primary (<5 yrs)

Note: Firm effects are normalized using the method in Card, Cardoso and Kline (2016).

Estimated Std. Study and country/industry Elasticity Error

Group 1: Industry-level profit measure1. Christofides and Oswald (1992), Canadian manufacturing 0.140 (0.035)2. Blanchflower, Oswald, Sanfey (1996), US manufacturing 0.060 (0.024)3. Estevao and Tevlin (2003), US manufacturing 0.290 (0.100)

Group 2: Firm-level profit measure, mean firm wage

4. Abowd and Lemieux (1993), Canadian manufacturing 0.220 (0.081)5. Van Reenen (1996), UK manufacturing 0.290 (0.089)6. Hildreth and Oswald (1997), UK 0.040 (0.010)7. Hildreth (1998), UK Manufacturing 0.030 (0.010)8. Barth et al (2016), US 0.160 (0.002)

Group 3: Firm-level profit measure, individual-specific wage

9. Margolis and Salvanes (2001), French manufacturing 0.062 (0.041)9. Margolis and Salvanes (2001), Norwegian manufacturing 0.024 (0.006)10. Arai (2003), Sweden 0.020 (0.004)11. Guiso, Pistaferri, Schivardi (2005), Italy 0.069 (0.025)12. Fakhfakh and FitzRoy (2004), French manufacturing 0.120 (0.045)13. Du Caju, Rycx, Tojerow (2011), Belgium 0.080 (0.010)14. Martins (2009), Portuguese manufacturing 0.039 (0.021)15. Guertzgen (2009), Germany 0.048 (0.002)16. Cardoso and Portela (2009), Portugal 0.092 (0.045)17. Arai and Hayman (2009), Sweden 0.068 (0.002)18. Card, Devicienti, Maida (2014), Italy (Veneto region) 0.073 (0.031)19. Carlsson, Messina, and Skans (2014), Swedish mfg. 0.149 (0.057)20. Card, Cardoso, Kline (2016), Portugal, between firm 0.156 (0.006)20. Card, Cardoso, Kline (2016), Portugal, within-job 0.049 (0.007)21. Bagger et al. (2014), Danish manufacturing 0.090 (0.020)22. Juhn et al. (2014), US manufacturing 0.020 (0.004)22. Juhn et al. (2014), US retail 0.037 (0.008)Note: see Appendix Table 1 for more complete description of each study.

Table 1: Summary of Estimated Rent Sharing Elasticities from the Recent Literature(Preferred specification, adjusted to TFP basis)

Table 2: Cross‐Sectional and Within‐Job Models of Rent Sharing for Portuguese Male Workers

Basic + major Basic + detailedBasic Specification industry/city industry/city

(1) (2) (3)

A. Cross Sectional Models (Worker‐year observations 2005‐2009)

1. OLS:  rent measure = mean log value added 0.270 0.241 0.207              per worker 2005‐2009 (0.017) (0.015) (0.011)

2. OLS:  rent measure = mean log sales per worker 0.153 0.171 0.159             2005‐2009 (0.009) (0.007) (0.004)

3. IV:  rent measure = mean log value added  0.327 0.324 0.292          per worker 2005‐2009.  Instrument = (0.014) (0.011) (0.008)          mean log sales per worker, 2004‐2010          First stage coefficient 0.475 0.541 0.562

[t=26.19] [t=40.72] [t=64.38]

B. Within‐Job Models (Change in Wages from 2005 to 2009 for stayers)

4. OLS:  rent measure = change in log value added 0.041 0.039 0.034              per worker from 2005 to 2009 (0.006) (0.005) (0.003)

5. OLS:  rent measure = change in log sales per 0.015 0.014 0.013             worker from 2005 to 2009 (0.005) (0.004) (0.003)

6. IV:  rent measure = change in log value added  0.061 0.059 0.056          per worker from 2005 to 2009.  Instrument = (0.018) (0.017) (0.016)          change in log sales per worker, 2004 to 2010          First stage coefficient 0.221 0.217 0.209

[t=11.82] [t=13.98] [t=18.63

Notes: Sample in panel A is 2,503,336 person‐year observations from QP for males working in 2005‐2009 between the ages of 19 and 65 with at least 2 years of potential experience employed at a firm with complete value added data (from SABI) for 2005 to 2009, and sales data (from QP) for 2004 and 2010.   Sample in panel B is 284,071 males age 19‐61 in 2005 who worked every year from 2005‐2009 at a firm with complete value added data (from SABI) for 2005 to 2009, and sales data (from QP) for 2004 and 2010. Standard errors clustered by firm ‐ 62,845 firms in panel A, 44,661 firms in panel B.  Models in panel A control for a cubic in experience and unrestricted education × year dummies.  Models in panel B control for a quadratic in experience and education.  Models in column 2 also control for 20 major industries and 2 major cities (Lisbon and Porto). Models in column 3 also control for 202 detailed industry dummies and 29 NUTS‐3 location dummies. 

 Table 3: Summary of Estimated AKM Models for Portuguese Men, Alternative Normalizations of Age Function

Baseline: Cubic Age Function Flat at Age 40

Cubic Age Function Flat at Age 50

Cubic Age Function Flat at Age 30

Cubic Age Function Flat at Age 0

Gaussian Basis Function

(1) (2) (3) (4) (5)

Std. dev. of person effects (across person‐yr obs.) 0.42 0.41 0.46 0.93 0.44Std. dev. of firm effects (across person‐yr obs.) 0.25 0.25 0.25 0.25 0.25Std. dev. of Xb (across person‐yr obs.) 0.07 0.10 0.12 0.74 0.08Correlation of person/firm effects 0.17 0.16 0.17 0.14 0.17Correlation of person effects and Xb 0.19 0.19 ‐0.32 ‐0.89 ‐0.06Correlation of firm effects and Xb 0.11 0.14 ‐0.03 ‐0.08 0.04

Inequality decomposition (Percent of variance of log wage explained):

   Person effects + Xb 63 63 63 63 63         Person effects 58 54 70 282 62         Xb 2 3 4 180 2         Cov. of person effects and Xb 3 5 ‐11 ‐399 ‐1

    Firm effects 20 20 20 20 20

    Cov. of firm effects with (person effect+Xb)  12 12 12 12 12          Cov. of firm effects with person effects 11 10 13 21 12          Cov. of firm effects with Xb 1 2 ‐1 ‐9 0

     Residual 5 5 5 5 5Notes: Sample includes 8,225,752 person‐year observations for male workers in largest connected set of QP in 2005‐2009 period.  Sample and baseline specification are the same as in Card, Cardoso and Kline (2016).   Models include 1,889,366 dummies for individual workers and 216,459 dummies for individual firms, year dummies interacted with education dummies, and function of age interacted with education dummies. Age function in models in columns 1‐4 includes quadratic and cubic terms, with age deviated from 40, 50, 30, and 0 for models in columns 1‐4, respectively.  Age function in model in column 5 is a Gaussian basis function with 5 equally spaced spline points.  All models have the same fit: RMSE of the model is 0.143, the adjusted R‐squared is 0.934.   

Table 4: Relationship Between Components of Wages and Mean Log Value Added per Worker

Basic + major Basic + detailedBasic Specification industry/city industry/city

(1) (2) (3)A. Combined Sample (n=2,252,436 person year observations at 41,120 firms)

1.  Log Hourly Wage 0.250 0.222 0.187(0.018) (0.016) (0.012)

2. Estimated Person Effect 0.107 0.093 0.074    (0.010) (0.009) (0.006)

3. Estimated Firm Effect 0.137 0.123 0.107(0.011) (0.009) (0.008)

4. Estimated Covariate Index 0.001 0.001 0.001(0.000) (0.000) (0.000)

B. Less‐Educated Workers (n=1,674,676 person year observations at 36,179 firms)

5.  Log Hourly Wage 0.239 0.211 0.181(0.017) (0.016) (0.011)

6. Estimated Person Effect 0.089 0.072 0.069    (0.009) (0.009) (0.005)

7. Estimated Firm Effect 0.144 0.133 0.107(0.015) (0.013) (0.008)

8. Estimated Covariate Index 0.000 0.000 0.000(0.000) (0.000) (0.000)

C. More‐Educated Workers (n=577,760 person year observations at 17,615 firms)

9.  Log Hourly Wage 0.275 0.247 0.196(0.024) (0.020) (0.017)

10. Estimated Person Effect 0.137 0.130 0.094(0.016) (0.013) (0.009)

11. Estimated Firm Effect 0.131 0.113 0.099(0.012) (0.009) (0.010)

12. Estimated Covariate Index ‐0.001 ‐0.001 ‐0.001(0.000) (0.000) (0.000)

Notes: Table entries are coefficients of mean log value added per worker (at current firm) in regression models with dependent variables listed in the row headings.  Standard errors clustered by firm in parentheses.  Sample in Panel B includes males with less than completed secondary education at firms in the connected set for less educated workers. Sample in Panel C includes males with high school education or more at firms in the connected set for more educated workers.  Sample in Panel A includes males in either the Panel B or Panel C sample. All models control for a cubic in experience and unrestricted education × year dummies.  Models in column 2 also control for 20 major industries and 2 major cities (Lisbon and Porto). Models in column 3 also control for 202 detailed industry dummies and 29 NUTS‐3 location dummies.

Appendix Table 1: Summary of Estimated Rent-Sharing Elasticities

Study Design Features Measure of Profitability Elasticity

A. Industry-Level Profit Measures

1. Christofides and Oswald (1992) Canadian union contracts; 120 narrowly Industry profits/worker 0.07defined manufacturing industries (wage changes)

2. Blanchflower, Oswald, and Sanfey (1996) US individual wage data (CPS), grouped to Industry profits/worker 0.01-0.06industry×year cells; manufacturing only (within-industry changes)

3. Estevao and Tevlin (2003) US manufacturing industry data; adjusted for Value added per worker 0.29labor quality; instrument for value-added = (first differences)demand shocks in downstream sectors

Profit per worker 0.14

B. Firm-Level Profit Measures, Average Firm-level Wages

(first differences)

4. Abowd and Lemieux (1993) Canadian union contracts merged to corporate Quasi-rent/worker 0.22accounts; instruments for revenues = industry (wage change model)selling prices, import and export prices

5. Van Reenen (1996) Large British manufacturing firms merged with Quasi-rent/worker 0.29corporate accounts; instruments for rents = (wage change model)innovations, imports, R&D, industry concentration

6. Hildreth and Oswald (1997) British firms (EXSTAT); firm-specific profits (from Profit per worker 0.02financial statements); instruments = lagged valuesof wages and profits

7. Hildreth (1998) British manufacturing establishments; Quasi-rent/worker 0.03establishment-specific value added;instruments for rents = innovation measure

8. Barth et al (2016) US establishments in LBD. Establishment- Sales/worker OLS = 0.32specific revenues; instrument for revenues/worker (within-establishment changes) IV = 0.16= revenues/worker in same industry, other regions

Note: Table continues.

Appendix Table 1 (continued): Summary of Estimated Rent-Sharing Elasticities

Study Design Features Measure of Profitability Elasticity

C. Individual Wages and Firm-Level Profit Measures

9. Margolis and Salvanes (2001) Worker and firm data for France and Norway; Profit per worker France: 0.03full time male workers in manufacturing; Norway: 0.01profit from financial filings; instruments= sales/worker and subsidies/worker

10. Arai (2003) Swedish worker panel matched to employer Change in 5-year average 0.01-0.02(10-year stayers design); profits from financial profit per workerstatements

11. Guiso, Pistaferri, and Schivardi (2005) Italian worker panel matched to larger firms; Permanent shock to log 0.07value added from financial statements; model- value added per workerbased decomposition of value added shocks Transitory shock to log 0.00

value added per worker

12. Fakhfakh and FitzRoy (2004) Larger French manufacturing establishments; Mean log value-added/worker 0.12value added from establishment survey over past 3 years

13. Du Caju, Rycx, and Tojerow (2011) Belgian establishment panel; value added and Value added minus labor 0.03-0.04labor cost from financial statements costs per worker

14. Martins (2009) Larger Portuguese manufacturing firms; Revenue-capital costs/worker 0.03-0.05revenue and capital costs from financial (differenced)statements; instruments=export share ofsales × exchange rate changes

15. Guertzgen (2009) German establishment/worker panel (LIAB) Quasi-rent/worker 0.03-0.04value added from establishment survey. (no adjustment for capital)instruments for change in quasi-rent = lags ofvalue added and wages Change in quasi-rent/worker 0.01-0.06

(stayers design)

Note: Table continues.

Appendix Table 1 (continued): Summary of Estimated Rent-Sharing Elasticities

Study Design Features Measure of Profitability Elasticity

C. Individual Wages and Firm-Level Profit Measures (continued)

16. Cardoso and Portela (2009) Portuguese worker panel; sales from firm reports; Permanent shock to log sales 0.09model-based decomposition of sales shocks

Transitory shock to log sales 0.00

17. Arai and Hayman (2009) Swedish worker/firm panel; profits from Change in profit per worker 0.07financial statements; stayers design;instrument=change in foreign sales

18. Card, Devicienti, and Maida (2014) Italian worker panel matched to firms; Value added per worker 0.06-0.08value added and capital from financial (within job match)statements; instrument for value added =sales/worker at firms in other regions

19. Carlsson, Messina, and Skans (2014) Swedish worker panel matched to firms; mining Firm-specfic output/worker 0.05and manufacturing only; firm-specific output (within-job-match)and selling price indexes; instruments forproductivity = indexes of firm-specific and Sectoral average output/worker 0.15sectoral TFPQ (within-job-match)

20. Card, Cardoso, and Kline (2016) Portuguese worker panel matched to firms; Mean value added per worker Males: 0.16value added and capital from financial Females: 0.14statements; wage measure=estimated firmeffect from AKM model Mean value added per worker Males: 0.05

(changes for stayers) Females: 0.04

21. Bagger, Christensen, and Mortensen (2014) Danish worker panel matched to firms; Output per worker Manuf: 0.09output from firm survey; non-parametric Trade: 0.13regressions within sector of wages on labor Transp/Comm: 0.05productivity Finance/Real Est: 0.07

22. Juhn, McCue, Monti, and Pierce (2016) US worker panel (LEHD) matched to firms; Changes in revenues per worker Manuf: 0.02output from firm surveys; stayers design for stayers over 3 years Retail: 0.04

Prof. Services: 0.04Finance: 0.01

Notes: estimates extracted by authors from studies listed.